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\begin{center}
\vskip 1cm{\LARGE\bf 
Space-Efficient Generation of  \\
\vskip .1in
Nonisomorphic Maps and Hypermaps
}
\vskip 1cm
\large
Timothy R. Walsh \\
Department of Computer Science\\
UQAM \\
Box 8888, Station A \\
Montr\'eal, Quebec, H3C 3P8 \\
Canada \\
\href{mailto:walsh.timothy@uqam.ca}{\tt walsh.timothy@uqam.ca}\\
\end{center}

\vskip .2 in

\begin{abstract}
In 1979, while working as a senior researcher in the Computing Centre of the
USSR Academy of Sciences in Moscow, I used Lehman's code for rooted maps of any
orientable genus to generate these maps.  By imposing an order on the code-words
and keeping only those that are maximal over all the words that code the same
map with each semi-edge chosen as the root, I generated these maps up to
orientation-preserving isomorphism, and by comparing each of them with the
code-words for the map obtained by reversing the orientation, I generated these
maps up to a generalized isomorphism that could be orientation-preserving or
orientation-reversing.  The limitations on the speed of the computer I was using
and the time allowed for a run restricted me to generating these maps with up to
only six edges.  In 2011, by optimizing the algorithms and using a more powerful
computer and more CPU time I was able to generate these maps with up to eleven
edges.  An average-case time-complexity analysis of the generation algorithms is
included in this article.  And now, by using a genus-preserving bijection
between hypermaps and bicoloured bipartite maps that I discovered in 1975 and the
condition on the word coding a rooted map for the map to be bipartite, I
generated hypermaps, both rooted and unrooted, with up to twelve darts
(edge-vertex incidence pairs).
\end{abstract}

\section{Introduction}
A \emph{map} is defined topologically as a 2-cell embedding~\cite{Arq87a} of a
connected graph, loops and multiple edges allowed, in a 2-dimensional surface. 
The \emph{faces} of a map are the connected components of the complement of the
graph in the surface. In this article the surface is assumed to be without
boundary and orientable, with an orientation already attributed to it
(counter-clockwise, say), so that it is completely described by a non-negative
integer $g$, its \emph{genus}.  For short, a map on a surface of genus $g$ will
be called a \emph{map of genus $g$}.  A \emph{planar map} is a map of genus 0 (a
map on a sphere).  If a map on a surface of genus $g$ has $v$ vertices, $e$
edges and $f$ faces, then by the Euler-Poincar\'e formula~\cite[Chap. 9]{Cox73}
\begin{equation}
f - e + v = 2(1-g).           \label{eq:1}
\end{equation}

Two maps are \emph{equivalent} if there is an orientation-preserving
homeomorphism between their embedding surfaces that takes the vertices, edges
and faces of one map into the vertices, edges and faces of the other.  A
\emph{dart} or \emph{semi-edge} of a map or graph is half an edge.  A loop is
assumed to be incident twice to the same vertex, so that every edge, whether or
not it is a loop, contains two darts.  The face incident to a dart $d$ is the
face incident to the edge containing $d$ and on the right of an observer on $d$
facing away from the vertex incident to $d$.  A \emph{rooted map} is a map with
a distinguished dart, its \emph{root}.  Two rooted maps are equivalent if there
is an orientation-preserving homeomorphism between their embedding surfaces that
takes the vertices, edges, faces and the root of one map into the vertices,
edges, faces and the root of the other.

A \emph{combinatorial map} is a connected graph with a cyclic order imposed on
the darts incident to each vertex, representing the order in which the darts of
a (topological) map are encountered during a rotation around the vertex
according to the orientation of the embedding surface.  Given a dart $d$, we
denote by $-d$ the other half of the edge containing $d$ and by $P(d)$ the next
dart after $d$ according to the cyclic order of the darts around the vertex
incident to $d$.  The darts incident to a face are encountered by successive
application of the permutation $P-$ (${-}~$Â&nbsp;followed by $P$).  In this way the
faces of a combinatorial map can be counted, so that its genus can be calculated
from~(\ref{eq:1}).  Two combinatorial maps are equivalent if they are related by
a \emph{map isomorphism} -- a graph isomorphism that preserves this cyclic order
-- with an analogous definition for the equivalence of two rooted combinatorial
maps. An \emph{automorphism} of a combinatorial map is a map isomorphism from a
map onto itself.

Following~\cite{Wormald81a} and~\cite{Wormald81b}, we define a \emph{sensed map}
to be an equivalence class of maps and an \emph{unsensed map} to be an
equivalence class of maps under a homeomorphism that could be
orientation-preserving or orientation-reversing.  It was shown in~\cite{JoSi}
that each equivalence class of topological maps is uniquely defined by an
equivalence class of combinatorial maps; so from now on a rooted map means a
rooted combinatorial map, a sensed map means an isomorphism class of
combinatorial maps and an unsensed map means an equivalence class of maps under
both isomorphism and reversal of the cyclic order imposed on the darts incident
to each vertex.

Map enumeration began in earnest with the work of Tutte, who used it in an
attempt to solve the famous four-colour problem.  Lehman used it in his study of
the molecular structure of polymers.  In addition, map enumeration has
applications in classical and algebraic combinatorics~\cite{GJ83}, theoretical
physics and integrable hierarchies~\cite{LZ04}.

There are many research papers on the enumeration of maps with various
properties; we list here some of the papers in which maps (rooted, sensed and
unsensed) have been enumerated by genus and either number of edges alone or
number of edges and vertices (the latter is equivalent by (\ref{eq:1}) to
enumerating by number of faces and vertices).

Rooted planar maps were counted by Tutte, first by number of edges alone (as a
closed-form formula)~\cite{Tu1} and then by number of faces and vertices (as a
generating function)~\cite{Tu3}.  I found an algorithm for counting rooted maps
by genus, number of edges and number of vertices~\cite{Walsh71,W1} and a
polynomial algorithm for counting rooted \emph{toroidal} maps (maps of genus 1),
both by number of edges and by number of vertices and faces~\cite{Walsh83}.
Using an improved version of the method of~\cite{Walsh71}, presented by Bender
and Canfield~\cite{BC86}, Arqu\`es found a closed-form formula for counting
rooted toroidal maps, both by number of edges and by number of vertices and
faces~\cite{Arq87a}.  Bender and Canfield found a closed-form formula for
counting rooted maps of genus 2 and 3 by number of edges~\cite{BC91}.
Giorgetti, a student of Arqu\`es, generalized the results of~\cite{Arq87a}
and~\cite{BC91} to obtain a general form for the generating function counting
rooted maps of any genus by number of vertices and faces and counted the maps of
genus 2 and 3~\cite{gio98a}.  I then collaborated with Giorgetti to extend this
enumeration up to genus 6~\cite{WG14} and later up to genus
10~\cite{WGMunrooted}.

Liskovets found a closed-form formula for the number of sensed planar maps by
number of edges~\cite{Liskovets81}.  Mednykh and Nedela generalized Liskovets'
method and thus counted sensed maps of genus 1, 2 and 3 by number of
edges~\cite{MN06} and then Giorgetti and Mednykh counted sensed maps of genus 4
by number of edges~\cite{MG11}. Then I collaborated with Giorgetti and Mednykh
to count sensed maps of genus up to 10 by number of vertices and faces and up to
genus 11 by number of edges~\cite{WGMunrooted,Walsh13}. 
Using a more efficient method for counting rooted maps discovered by Carrell and
Chapuy~\cite{CC14-v3}, Giorgetti and I enumerated rooted and sensed maps of
genus up to 50 with up to 100 edges in~\cite{GW14}, which includes tables of numbers
of sensed maps of genus up to 19.
 And Wormald found an
algorithm for counting planar maps, both sensed and unsensed, by number of edges
and by number of vertices and faces~\cite{Wormald81a,Wormald81b}.  The methods
used to obtain all of the above results are computationally more efficient than
exhaustive generation.  But, as far as I know, exhaustive generation is the only
method yet known to enumerate unsensed non-planar maps, and even for maps that
have been enumerated by other methods, exhaustive generation serves to verify
the numbers obtained by these methods.

The method I used in~\cite{Walsh83b} to generate isomorphism classes of maps
without having to store all the previously generated rooted maps to see whether
each new map is isomorphic to one of the old ones is essentially the one used by
Read~\cite{Read78} to generate the isomorphism classes of 9â€‘-vertex graphs.  He
generated the adjacency matrix of each of the labelled 9-vertex graphs and then
eliminated all those that are not lexicographically largest among those matrices
representing the same graph but with a different labelling.  Since a rooted map
has only the trivial automorphism~\cite{Tu1}, I generated all the rooted maps,
or rather Lehman's code for rooted maps, with $e$ edges and $v$ vertices,
eliminated all those whose code-word is not lexicographically largest among
those coding the same map but with a different root, and sorted the rest by
genus to generate sensed maps; to generate unsensed maps, I eliminated each
sensed map whose code-word could be made lexicographically larger by  reversing
the cyclic order of the darts at each vertex and choosing one of the darts as
the root.  To be sure, more sophisticated methods of generating isomorphism
classes of combinatorial objects have since been discovered~\cite{GLM97}, and
for objects with many distinct labellings these methods are probably much
faster.  However, a map with $e$ edges has at most $2e$ distinct rootings; so
the admittedly old-fashioned method I used seems to be quite adequate.

More recently Jackson and Visentin published an atlas of maps~\cite{JV01}.

A (combinatorial) \emph{hypermap} is a generalization of a map in which an edge
is allowed to have any positive number of darts instead of exactly two and the
darts are cyclically ordered around the edges as well as the vertices.  In 1975
I published a genus-preserving bijection between hypermaps with $d$ darts, $e$
edges, $v$ vertices and $f$ faces and bicoloured bipartite maps with $d$ edges,
$e$ black vertices, $v$ white vertices and $f$ faces, each containing twice as
many darts as the corresponding face of the hypermap~\cite{Walsh75}.  This
bijection was used by Arqu\`es to count rooted planar~\cite{Arq86a} and
toroidal~\cite{Arq87} hypermaps by number of vertices, edges and faces;
Chauve~\cite{Chauve03} independently counted rooted bicoloured bipartite planar
maps with the corresponding parameters.  And now I discovered a condition on the
Lehman word that codes a rooted map for the map to be bipartite, which I used to
generate rooted, sensed and unsensed hypermaps with up to 12 darts.

The words with which Lehman coded rooted maps are described in
Section~\ref{sec2}, the procedure I used to generate these words is described
and analyzed in Section~\ref{sec3}, a discussion of the generation of hypermaps
appears in Section~\ref{sec4} and the results of the computation, including
timings, are described in Section~\ref{sec5}.  A table of numbers of unsensed
maps with up to 11 edges, sorted by genus and number of vertices, appears in
Appendix~\ref{appA}; the analogous tables for rooted maps and sensed maps
appear in other sources, which are cited in Section~\ref{sec5}. 
Appendix~\ref{appB} contains a table of numbers of rooted, sensed and unsensed
hypermaps.

\section{Lehman's code for rooted maps}
\label{sec2}

In the 1960s Lehman, who was then my Ph.~D. supervisor, generalized the code for
a rooted plane tree as a balanced parenthesis system to a code for a rooted
planar map with a given spanning tree as a (balanced) parenthesis system (coding
the rooted plane tree obtained by deleting the edges not in the spanning tree)
merged with a bracket system (coding the rooted one-vertex map obtained by
contracting the edges of the spanning tree).  The number of pairs of parentheses
is the number of edges of the spanning tree and the number of pairs of brackets
is the number of edges not in the spanning tree.  To code a rooted planar map
without a spanning tree, he used Tamari's maze-running algorithm~\cite{Tarry95},
which is essentially depth-first search~\cite{Tarjan72} with the darts incident
to each vertex encountered in their cyclic order, to construct a canonical
spanning tree, and he proved that a spanning tree is canonical if and only if
the code word for the rooted map with this spanning tree does not contain the
forbidden sub-word [(]), where the right bracket is the \emph{mate} of (that is,
closes) the left bracket, the right parenthesis is the mate of the left
parenthesis and the four symbols are not necessarily contiguous.

To code a rooted map of any orientable genus, he replaced the bracket system by
an \emph{integer system on $m$ pairs}: a word consisting of two copies of each
of the integers 1, 2, \ldots, $m$, where $m$ is the number of edges in the
rooted one-vertex map coded by the integer system and the first occurrences of
the integers are in increasing order.  The forbidden sub-word is now $i(i)$,
where the right parenthesis is the mate the left one.

Each letter in a word coding a rooted map represents a dart, with the first
letter representing the root.  If a dart $d$ is (represented by) a parenthesis
or a bracket, then $-d$ is its mate; if $d$ is an integer $i$, then $-d$ is the
other occurrence of $i$.  If $d$ is a bracket or an integer, then $P(d)$ is the
next letter in the word (with wraparound); if $d$ is a parenthesis, then $P(d)$
is the letter that follows the mate of $d$ (with wraparound).  The darts of the
face containing $d$ can be found from the code-word by successive application of
the permutation $P-$ to the letters representing the darts.  For example, in the
code word 123123, the face containing the first 1 also contains the second 2 and
the first 3 (the next dart would be the first 1) and the face containing the
first 2 also contains the second 3 and the second 1; since all the letters
belong to one of these two faces, there are only 2 faces and so by (\ref{eq:1})
the one-vertex map coded by this word is of genus 1.  Since contracting an edge
does not change the genus of a map, the genus of a rooted map can be calculated
from the integer sub-system of its code-word.


A more detailed description of Professor Lehman's code, including his method
of coding a rooted map, can be found in my Ph.~D. thesis~\cite{Walsh71} and
in~\cite{Walsh83b}, where I described the use I made of his code to generate
isomorphism classes of maps.


\section{Generating maps}
\label{sec3}

To generate the rooted plane trees with $e$ edges, I generate the parenthesis
systems with $e$ pairs of parentheses in lexicographical order, with a left
parenthesis represented by 0 and a right parenthesis represented by -1.   To
generate the rooted planar one-vertex maps with $e$ edges, I generate the
bracket systems with $e$ pairs of brackets, also in lexicographical order, with
a left bracket represented by 2 and a right bracket represented by 1.  To
generate the not-necessarily-planar rooted one-vertex maps with $e$ edges, I
generate the integer systems on $e$ pairs; in~\cite{Walsh83b} I made the second
occurrence of each integer move from its leftmost position (immediately to the
right of the first occurrence of the same integer) to its rightmost position
(the rightmost letter in the word) with $e$ moving the fastest, whereas now I
use a Gray code in which they move alternately to the right and to the left. 
Each new system is generated in $O(e)$ time in the worst case and $O(1)$ time in
the average case.

To generate the rooted maps with $e$ edges and $v$ vertices, I first generate
the bracket systems or the integer systems on e-v+1 pairs, and in the latter
case I calculate the genus by counting the faces (in $O(e)$ time) and
substituting into (\ref{eq:1}) as described above.  For each bracket system or
integer system I generate all the parenthesis systems on $v-1$ pairs.  For each
pair of words I merge them in all possible ways that avoid the forbidden
sub-word, moving each parenthesis from left to right, with a right parenthesis
starting adjacent to its mate and stopping when it hits an integer or bracket
whose mate is to the left of the parenthesis' mate or when it passes the last
integer or bracket.  The procedure for passing from one merged word to the next
is described in more detail in~\cite{Walsh83b}.  This procedure involves
deleting a parenthesis when it reaches its rightmost position and then, when a
parenthesis has been moved to the right, inserting all the deleted parentheses
in their leftmost positions.  Since in the worst case all the parentheses may
get deleted and reinserted in passing from one word to the next, the algorithm
runs in $O(e^2)$ worst-case time if the letters following a deleted parenthesis
are pulled to the left as in~\cite{Walsh83b}.  Now I replace each deleted
parenthesis by a marker ($-$2).  After a parenthesis has been moved to the
right, some of the slots between successive undeleted parentheses (or to the
left of the leftmost parentheses or to the right of the rightmost one) will
contain both markers and either integers or brackets.  In each such slot I move
all the markers to the left side of the slot and all the integers or brackets to
the right side and then replace all the markers by the deleted parentheses, so
that the algorithm runs in $O(e)$ worst-case time.

To generate the sensed maps with $e$ edges and $v$ vertices, I generate the
rooted maps with $e$ edges and $v$ vertices, or rather, their Lehman code-words,
and then I check each one for lexicographical maximality with respect to the
code-words for all the rootings of the same map.  To this end, I decode the
code-word into a rooted map represented by two arrays VERT and NEXT, where the
darts are the indices $1, 2, \ldots, 2e$, the $i$th edge encountered during the
decoding procedure consists of the darts $i \leq e$ and $2e+1-i$, VERT[$i$] is
the label assigned to the vertex containing the dart $i$, NEXT[$i$] is $P(i)$
and the root is dart 1.  Then, I code this map rooted at each of the other darts
and compare the new code-word with the original one.  Of course, it is not
usually necessary to try every dart or even to complete each coding procedure. 
Since the order is lexicographical, as soon as a letter in the new code-word
differs from a letter in the same position in the old one I can terminate the
coding; if the new letter is bigger, the old code-word is not maximal and I
reject it, and if the new letter is smaller, I try the next dart.  If all the
darts have been tried and the old code-word hasn't been rejected, I accept
(count) it as the representative rooted map of a sensed map.

The decoding and coding procedures each run in $O(e)$ time so that the testing
procedure runs in $O(e^2)$ time in the worst case: if the map has $2e$
automorphisms, then all the $2e$ codes for this map are identical, so that all
the darts must be tried and each code-word must be constructed in its entirety. 
But, as we will show, the average time for the testing procedure is $O(e \ln
e)$.

Almost all maps have only the trivial automorphism~\cite{RW95}; so in almost all
cases the $2e$ words that code the same map rooted at each of the darts will be
distinct.  If the old code-word is the ith smallest one among those $2e$ words,
this process can be modelled by removing balls at random without replacement
from an urn containing $i-1$ black balls (words smaller than the old one) and
$2e-i$ white ones (words bigger than the old one) until either a white ball is
removed or all the balls are removed.  If instead the black balls are replaced,
the probability that the next ball will be white decreases, so that the expected
number of removals increases.  The upper bound thus obtained for the expect
number of removals is easy to calculate: it is

\begin{equation}
\label{eq2}
p + 2 (1-p) p + 3 (1-p)^2 p + \ldots = 1/p,
\end{equation}

where $p$ is the probability of removing a white ball, which is $(2e-i)/(2e-1)$.
If $i=2e$, (\ref{eq2}) is not defined, but in this case (when all the balls are
black because the old code-word is maximal) (\ref{eq2}) is replaced by the
number of removals without replacement, which is $2e$.  Since the words coding a
given map rooted at all its darts will be generated, $i$ will assume all the
values $1,Â&nbsp;2,Â&nbsp;\ldots,Â&nbsp;2e$; so the sum of the expected values is less than
\begin{equation}
\label{eq3}
2e + (2e-1) \, \left[1/1+1/2+1/3+\ldots+1/(2e-1)\right],
\end{equation}

which is asymptotic to $2 e \ln e$.  The expected number of darts that have to
be tried for each generated code word is thus $O(\ln e)$.

To estimate the cost of comparing an old code-word with a new one, let $i$ be
the smallest index of a letter in which the new code-word differs from the old
one. The expected value of $i$ is given by (\ref{eq2}), where $p$ is now the
probability that two letters chosen at random from an alphabet are distinct,
which is equal to $(aâ€‘1)/a$, where $a$ is the number of letters in the alphabet. 
Since the alphabet has at least two letters if $e &gt; 1$, in the average case the
number of letters of the new code-word that have to be constructed is bounded by
a constant.  However, each coding begins by initializing all the vertices to
``new'', which takes $O(e)$ time; so the average time for testing a code-word
for maximality is $O(e \ln e)$.  The testing procedure is shown in
Figure~\ref{fig1}.

To generate the unsensed maps with $e$ edges and $v$ vertices I generate the
sensed ones and then, for each sensed map (which I have already constructed by
decoding), I reverse the cyclic order of the darts incident to each vertex by
constructing the array PREV, where PREV[NEXT[$i$]] = $i$ for each $i$.  This
step runs in $O(e)$ time.  Then I code the reversed map at every dart and
compare the new code-word with the old maximal code-word (with the same
shortcut, and thus the same average-case time-complexity) and accept the old
code-word as the representative rooted map of an unsensed map if none of the
comparisons have rejected it.

\begin{figure}[htb!]
\begin{small}
Procedure IsMax ($W$, a code-word of length $2e$ with $p$ parenthesis pairs)
\begin{enumerate}
 \item 
 Decode $W$ into a map $M$ with $e$ edges and $p + 1$ vertices rooted at
 dart 1 \\
 // $O(e)$ time and space;

\item Calculate the genus $g$ of $M$; // $O(e)$ time

\item Set Maxword to True;

\item For $d$ from 2 to $2e$  // $d$ is the current dart

\item 
$\ $ Initialize the coding of $M$ rooted at $d$ by setting all
the vertices to new; // $O(e)$ time

\item $\ $ For $i$ from 1 to $2e$

\item $\ \ $ Set $X[i]$ to the $i$th letter of the word that
codes $M$ rooted at $d$; // $O(1)$ time

\item $\ \ $ If $X[i] &gt; W[i]$ then set Maxword to False and
exit loop; // $W &lt; X$; so $W$ is not maximal.

\item $\ \ $ If $X[i] &lt; W[i]$ then exit loop; // $X &lt; W$; so this dart
 need no longer be used as a root.

\item $\ $ End for $i$; // $O(e)$ worst case and $O(1)$ average-case time

\item $\ $ If Maxword = False then exit loop; // $W$ is not maximal;
so it will be rejected.

\item End for $d$; // $O(e)$ worst-case and $O(\ln e)$ average-case number of
    iterations \\
 // $O(e^2)$ worst-case and $O(e \ln e)$ average-case time to test $W$ for
    maximality

\item If Maxword = True then // $W$ is maximal; so it is chosen as the
    representative of $M$.

$\ $ increase by 1 the number of sensed genus-$g$ maps with $e$ edges and
$p+1$ vertices;
\end{enumerate}
End IsMax.
\end{small}
\caption{The algorithm for testing whether a code-word represents a
sensed map.\label{fig1}}
\end{figure}


\begin{figure}[htb!]
\begin{tabular}{ccc} 
\begin{minipage}{0.45\textwidth}
\begin{center}
\begin{tikzpicture}[scale=.4,line cap=round,line join=round,x=1.0cm,y=1.0cm]
 \node[draw,circle] (v1) at (9,9) {$1$};
 \draw (10,10.6) node {$1$};
 \draw (10,7.4) node {$8$};
 \draw (8.2,10.6) node {$2$};
 \draw (8.2,7.4) node {$3$};
 \node[draw,circle] (v2) at (5,13) {$2$};
 \draw (5.6,11.4) node {$7$}; 
 \node[draw,circle] (v3) at (5,5) {$3$};
 \draw (5.6,6.6) node {$6$};
 \node[draw,circle] (v4) at (1,1) {$4$};
 \draw (4.2,3.4) node {$4$};
 \draw (1.6,2.6) node {$5$};
 \draw[thick] (v1) to (v2);
 \draw[thick] (v1) to (v3);
 \draw[thick] (v3) to (v4);
  \draw[thick] (v1) to [curve
    to,looseness=16,in=-40,out=40] (v1);
\end{tikzpicture}
\end{center}
\end{minipage}
&amp; &amp;
\begin{minipage}{0.45\textwidth}
\begin{tabular}{lllllllll}
index $i$: &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 &amp; 7 &amp; 8 \\
VERT[$i$]: &amp; 1 &amp; 1 &amp; 1 &amp; 3 &amp; 4 &amp; 3 &amp; 2 &amp; 1 \\
NEXT[$i$]: &amp; 2 &amp; 3 &amp; 8 &amp; 6 &amp; 5 &amp; 4 &amp; 7 &amp; 1 \\
PREV[$i$]: &amp; 8 &amp; 1 &amp; 2 &amp; 6 &amp; 5 &amp; 4 &amp; 7 &amp; 3
\end{tabular} 
\end{minipage}
\end{tabular}
\caption{The planar map rooted at dart 1 coded by [ ( ) ( ( ) ) ]. \label{fig2}}
\end{figure}

For example, at some point during the generation of the rooted planar maps with
4 edges and 4 vertices the word [ ( ) ( ( ) ) ] will be generated.  This is the
Lehman code-word for the rooted map drawn on the left side of Figure~\ref{fig2},
where the darts and the vertices are labelled in the order in which the edges
and vertices are encountered during the decoding procedure (dart 1 is the root).
 The arrays VERT, NEXT and PREV are shown on the right side of Figure~\ref{fig2}.

When this map is coded using any of the darts 2, \ldots, 7 as the root, the first
letter is (, which is represented by 0, whereas the first letter of the old
code-word is $[$, represented by 2; so the coding terminates immediately. 
However, when dart 8 is used as the root, the first two letters are [ ].  The
second letter of the new code-word is represented by 1, whereas the second
letter of the old code-word is represented by 0; so the old code-word is not
maximal and is rejected.

Later during the generation of the same set of rooted planar maps the word
[~]~(~)~(~(~)~) will be generated.  This word codes the same map rooted at dart
8.
All of the other darts will yield a lexicographically smaller code; so this word
will be accepted as the representative of the map drawn in Figure~\ref{fig2} as
a sensed map.  But when the cyclic orders are reversed and the dart labelled 1
in the diagram is used as the root, the code-word is [~]~(~(~)~)~(~) .  This
word first differs from the previous one in the fourth letter, which is
represented by 0 in the new word and by â€“1 in the old word; so the old word will
be rejected as an unsensed map as soon as the fourth letter has been computed. 
But when the new word is generated, it will be accepted as both a sensed map and
an unsensed map; so this map will count as two sensed maps and one unsensed map.

\section{Generating hypermaps}
\label{sec4}

To generate rooted hypermaps it suffices to generate bicoloured bipartite maps
rooted at an edge or, equivalently, rooted at a dart that is incident to a white
vertex.  This was done by using the following theorem.

\begin{theorem}
A rooted map is bipartite if and only if its code-word has the property
that between every pair of matching brackets or integers there are an odd number
of parentheses.
\end{theorem}

\begin{proof}
The spanning tree coded by the parenthesis sub-word is bicoloured, with
the vertex incident to the root coloured white.  A pair of matching brackets or
integers is written when the two darts of an edge $e$ that is not in the spanning
tree are encountered during the coding process.  Each parenthesis between the
members of the pair is written when an edge in the spanning tree is traversed,
thus passing from a vertex of one colour to a vertex of the other colour.  The
two darts of $e$ are thus incident to vertices of opposite colours if and only if
the number of parentheses between the matching brackets or integers is odd.  If
this condition holds for every pair of matching brackets or integers, then the
map is properly coloured in two colours and is thus bipartite.  If this
condition is violated for at least one matching pair, then the map is not
properly coloured in two colours, and since the colouring of the spanning tree
is uniquely determined by the colour of the vertex containing the root, the map
cannot be properly coloured in two colours and is thus not bipartite.  This
completes the proof.
\end{proof}

I modified the program to generate just those code words that both avoid the
forbidden sub-word and satisfy the condition stated in the theorem, so that it
generates the words coding the rooted bipartite maps that are in bijection with
hypermaps of the same genus.  To this end, I move brackets or integers from
right to left, separating the two members of each pair of brackets or integers
by an odd number of parentheses, instead of moving parentheses from left to
right as in~\cite{Walsh83b}.  To test a code word for maximality, I compare it
with all the words coding the same map but with a different root incident to a
white vertex, and then with all the words coding the orientation-reversed map
with any root incident to a white vertex.  In this way I generated all the
hypermaps -- rooted, sensed, and unsensed -- with up to 12 darts.

The time-complexity of the algorithm for generating hypermaps is the same as
for maps. Since only the old and the new word have to be stored at any one time
and each word is only $O(e)$ letters long, the space-complexity of the
generation algorithm is $O(e)$ for both maps and hypermaps.  Counting the words
and sorting the numbers by genus and the other parameters takes $O(e)$ space for
planar maps, $O(e^2)$ space for planar hypermaps and maps that are not
necessarily planar, and $O(e^3)$ space for hypermaps that are not necessarily
planar.

\section{The results of the computation}
\label{sec5}

The work described in~\cite{Walsh83b} was done in 1979 in the Computing Centre
of the USSR Academy of Sciences in Moscow on a BESM-6 computer, which has a 10
megahertz clock speed, and users were restricted to 5 minutes of CPU time per
run.  Within these limitations I was able to do the
calculations for maps with up to only 6 edges, processing a total of 110,410
6-edge rooted maps.  I published these results, including a table of numbers of
sensed and unsensed maps, in~\cite{Walsh83b}.  In 2011, using my Macbook Pro
laptop, which has a duo processor and a 2.66 gigahertz clock speed, being
subject to no run time restrictions, programming in C instead of FORTRAN and
optimizing the algorithms, I was able to extend the calculations up to 11 edges;
the run time for 11 edges, which processes 285,764,591,114 rooted maps, was
about a week.  For 10 edges it was about a day, for 9 edges about three hours, for 8
edges about 20 minutes, for 7 edges about 2 minutes, for 6 edges about 10
seconds and for fewer than 6 edges it was too short to be measured.  In each
case the time was roughly proportional to the number of rooted maps, verifying
experimentally the above average-case time complexity for maximality testing. 
Further verification was provided by the following time trial: it took two
minutes to generate all the rooted planar maps with 10 edges and less than six
minutes to generate all the unsensed planar maps with 10 edges.  For unsensed
hypermaps, the computation time was 8 seconds for 9 darts, 2 minutes for 10
darts, 33 minutes for 11 darts and about 10 hours (to process 5,201,061,455
rooted bipartite maps) for 12 darts.

The numbers of rooted maps generated by my program agree with the tables in my
joint paper with Prof. Lehman~\cite{W1}; these tables go up to 11 edges, and the
tables in~\cite{Walsh71} go up to 14 edges.  The numbers of sensed maps agree
with the numbers calculated jointly with Giorgetti and Mednykh without
generating maps; tables for the non-planar maps with up to 11 edges appear
in~\cite{WGMunrooted} and~\cite{Walsh13}.  The numbers of unsensed maps with up
to 6 edges agree with the tables in~\cite{Walsh83b}.   The numbers of unsensed
planar maps agree with those in unpublished tables given to me by Wormald, who
counted those maps and published his results in~\cite{Wormald81a}
and~\cite{Wormald81b}.  The numbers of rooted and sensed hypermaps of genus 0
and 1 with $d$ darts agree with those published by Mednykh and 
Nedela~\cite{MN10}.  The numbers of rooted hypermaps of genus 0 and 1, counted
by number of vertices, edges and faces, agree with those published by
Chauve~\cite{Chauve03} and Arqu\`es~\cite{Arq87}, respectively.  For unsensed
non-planar maps with more than 6 edges, as well as for all the other types of
hypermaps, the numbers I generated are, as far as I know, new.

The source code is available as a text file~\cite{WalshCode}.  It will run on
any 64-bit
computer that runs C programs.

\section{Acknowledgment} I wish to thank NSERC for partially supporting this
research, and Alain Giorgetti and Alexander Mednykh for suggestions for
improving the presentation of this article.

\appendix
\subsubsection*{Appendix A: The number of unsensed genus-g maps with e edges and v vertices.}
\label{appA}


\begin{scriptsize}
\begin{verbatim}
  E   v        g=0        g=1        g=2        g=3        g=4        g=5       all g

  0   1          1                                                                  1
  0 sum          1                                                                  1

  1   1          1                                                                  1
  1   2          1                                                                  1
  1 sum          2                                                                  2

  2   1          1          1                                                       2
  2   2          2          0                                                       2
  2   3          1          0                                                       1
  2 sum          4          1                                                       5

  3   1          2          3                                                       5
  3   2          5          3                                                       8
  3   3          5          0                                                       5
  3   4          2          0                                                       2
  3 sum         14          6                                                      20

  4   1          3         10          4                                           17
  4   2         13         20          0                                           33
  4   3         20         10          0                                           30
  4   4         13          0          0                                           13
  4   5          3          0          0                                            3
  4 sum         52         40          4                                           96

  5   1          6         35         38                                           79
  5   2         35        125         38                                          198
  5   3         83        125          0                                          208
  5   4         83         35          0                                          118
  5   5         35          0          0                                           35
  5   6          6          0          0                                            6
  5 sum        248        320         76                                          644

  6   1         12        132        328         82                               554
  6   2        104        728        739          0                              1571
  6   3        340       1226        328          0                              1894
  6   4        504        728          0          0                              1232
  6   5        340        132          0          0                               472
  6   6        104          0          0          0                               104
  6   7         12          0          0          0                                12
  6 sum       1416       2946       1395         82                              5839

  7   1         27        513       2569       2174                              5283
  7   2        315       4036       9906       2174                             16431
  7   3       1401      10133       9906          0                             21440
  7   4       2843      10133       2569          0                             15545
  7   5       2843       4036          0          0                              6879
  7   6       1401        513          0          0                              1914
  7   7        315          0          0          0                               315
  7   8         27          0          0          0                                27
  7 sum       9172      29364      24950       4348                             67834

  8   1         65       2072      18512      37439       7258                  65346
  8   2       1021      21733     105905      85172          0                 213831
  8   3       5809      75202     178502      37439          0                 296952
  8   4      15578     111544     105905          0          0                 233027
  8   5      21420      75202      18512          0          0                 115134
  8   6      15578      21733          0          0          0                  37311
  8   7       5809       2072          0          0          0                   7881
  8   8       1021          0          0          0          0                   1021
  8   9         65          0          0          0          0                     65
  8 sum      66366     309558     427336     160050       7258                 970568

  9   1        175       8558     124044     488891     344488                 966156
  9   2       3407     113721     967844    1859361     344488                3288821
  9   3      24299     514014    2401662    1859361          0                4799336
  9   4      82546    1046261    2401662     488891          0                4019360
  9   5     149007    1046261     967844          0          0                2163112
  9   6     149007     514014     124044          0          0                 787065
  9   7      82546     113721          0          0          0                 196267
  9   8      24299       8558          0          0          0                  32857
  9   9       3407          0          0          0          0                   3407
  9  10        175          0          0          0          0                    175
  9 sum     518868    3365108    6987100    4696504     688976               16256556

 10   1        490      35655     781919    5293283    8808724    1491629    16411700
 10   2      11814     580810    7887415   29372094   19848849          0    57700982
 10   3     102010    3294692   26625471   49022864    8808724          0    87853761
 10   4     426879    8728573   39172217   29372094          0          0    77699763
 10   5     972660   11966785   26625471    5293283          0          0    44858199
 10   6    1273644    8728573    7887415          0          0          0    17889632
 10   7     972660    3294692     781919          0          0          0     5049271
 10   8     426879     580810          0          0          0          0     1007689
 10   9     102010      35655          0          0          0          0      137665
 10  10      11814          0          0          0          0          0       11814
 10  11        490          0          0          0          0          0         490
 10 sum    4301350   37246245  109761827  118353618   37466297    1491629   308620966

 11   1       1473     149257    4690016   50026987  159968175   97864389   312700297
 11   2      41893    2901436   58891739  374871812  596357213   97864389  1130928482
 11   3     429509   20057276  256786053  912749995  596357213          0  1786380046
 11   4    2158241   66570286  513820635  912749995  159968175          0  1655267332
 11   5    6030752  118697249  513820635  374871812          0          0  1013420448
 11   6    9953314  118697249  256786053   50026987          0          0   435463603
 11   7    9953314   66570286   58891739          0          0          0   135415339
 11   8    6030752   20057276    4690016          0          0          0    30778044
 11   9    2158241    2901436          0          0          0          0     5059677
 11  10     429509     149257          0          0          0          0      578766
 11  11      41893          0          0          0          0          0       41893
 11  12       1473          0          0          0          0          0        1473
 11 sum   37230364  416751008 1668376886 2675297588 1512650776  195728778  6506035400
\end{verbatim}
\end{scriptsize}

\subsubsection*{Appendix B: The number of hypermaps of genus g.}
\label{appB}

\begin{scriptsize}
\begin{verbatim}
Number of rooted hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0       all g
  1   1   1           1           1
  1     sum           1           1

Number of sensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0       all g
  1   1   1           1           1
  1     sum           1           1

Number of unsensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0       all g
  1   1   1           1           1
  1     sum           1           1


Number of rooted hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0       all g
  2   1   1           1           1
  2   1   2           1           1
  2   2   1           1           1
  2     sum           3           3

Number of sensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0       all g
  2   1   1           1           1
  2   1   2           1           1
  2   2   1           1           1
  2     sum           3           3

Number of unsensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0       all g
  2   1   1           1           1
  2   1   2           1           1
  2   2   1           1           1
  2     sum           3           3


Number of rooted hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1       all g
  3   1   1           1           1           2
  3   1   2           3           0           3
  3   2   1           3           0           3
  3   1   3           1           0           1
  3   2   2           3           0           3
  3   3   1           1           0           1
  3     sum          12           1          13

Number of sensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1       all g
  3   1   1           1           1           2
  3   1   2           1           0           1
  3   2   1           1           0           1
  3   1   3           1           0           1
  3   2   2           1           0           1
  3   3   1           1           0           1
  3     sum           6           1           7

Number of unsensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1       all g
  3   1   1           1           1           2
  3   1   2           1           0           1
  3   2   1           1           0           1
  3   1   3           1           0           1
  3   2   2           1           0           1
  3   3   1           1           0           1
  3     sum           6           1           7


Number of rooted hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1       all g
  4   1   1           1           5           6
  4   1   2           6           5          11
  4   2   1           6           5          11
  4   1   3           6           0           6
  4   2   2          17           0          17
  4   3   1           6           0           6
  4   1   4           1           0           1
  4   2   3           6           0           6
  4   3   2           6           0           6
  4   4   1           1           0           1
  4     sum          56          15          71

Number of sensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1       all g
  4   1   1           1           2           3
  4   1   2           2           2           4
  4   2   1           2           2           4
  4   1   3           2           0           2
  4   2   2           5           0           5
  4   3   1           2           0           2
  4   1   4           1           0           1
  4   2   3           2           0           2
  4   3   2           2           0           2
  4   4   1           1           0           1
  4     sum          20           6          26
\end{verbatim}
\end{scriptsize}
\newpage
\begin{scriptsize}
\begin{verbatim}
Number of unsensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1       all g
  4   1   1           1           2           3
  4   1   2           2           2           4
  4   2   1           2           2           4
  4   1   3           2           0           2
  4   2   2           5           0           5
  4   3   1           2           0           2
  4   1   4           1           0           1
  4   2   3           2           0           2
  4   3   2           2           0           2
  4   4   1           1           0           1
  4     sum          20           6          26


Number of rooted hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2       all g
  5   1   1           1          15           8          24
  5   1   2          10          40           0          50
  5   2   1          10          40           0          50
  5   1   3          20          15           0          35
  5   2   2          55          40           0          95
  5   3   1          20          15           0          35
  5   1   4          10           0           0          10
  5   2   3          55           0           0          55
  5   3   2          55           0           0          55
  5   4   1          10           0           0          10
  5   1   5           1           0           0           1
  5   2   4          10           0           0          10
  5   3   3          20           0           0          20
  5   4   2          10           0           0          10
  5   5   1           1           0           0           1
  5     sum         288         165           8         461

Number of sensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2       all g
  5   1   1           1           3           4           8
  5   1   2           2           8           0          10
  5   2   1           2           8           0          10
  5   1   3           4           3           0           7
  5   2   2          11           8           0          19
  5   3   1           4           3           0           7
  5   1   4           2           0           0           2
  5   2   3          11           0           0          11
  5   3   2          11           0           0          11
  5   4   1           2           0           0           2
  5   1   5           1           0           0           1
  5   2   4           2           0           0           2
  5   3   3           4           0           0           4
  5   4   2           2           0           0           2
  5   5   1           1           0           0           1
  5     sum          60          33           4          97

Number of unsensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2       all g
  5   1   1           1           3           4           8
  5   1   2           2           7           0           9
  5   2   1           2           7           0           9
  5   1   3           4           3           0           7
  5   2   2          10           7           0          17
  5   3   1           4           3           0           7
  5   1   4           2           0           0           2
  5   2   3          10           0           0          10
  5   3   2          10           0           0          10
  5   4   1           2           0           0           2
  5   1   5           1           0           0           1
  5   2   4           2           0           0           2
  5   3   3           4           0           0           4
  5   4   2           2           0           0           2
  5   5   1           1           0           0           1
  5     sum          57          30           4          91


Number of rooted hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2       all g
  6   1   1           1          35          84         120
  6   1   2          15         175          84         274
  6   2   1          15         175          84         274
  6   1   3          50         175           0         225
  6   2   2         135         456           0         591
  6   3   1          50         175           0         225
  6   1   4          50          35           0          85
  6   2   3         262         175           0         437
  6   3   2         262         175           0         437
  6   4   1          50          35           0          85
  6   1   5          15           0           0          15
  6   2   4         135           0           0         135
  6   3   3         262           0           0         262
  6   4   2         135           0           0         135
  6   5   1          15           0           0          15
  6   1   6           1           0           0           1
  6   2   5          15           0           0          15
  6   3   4          50           0           0          50
  6   4   3          50           0           0          50
  6   5   2          15           0           0          15
  6   6   1           1           0           0           1
  6     sum        1584        1611         252        3447

Number of sensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2       all g
  6   1   1           1           7          16          24
  6   1   2           3          31          16          50
  6   2   1           3          31          16          50
  6   1   3          10          31           0          41
  6   2   2          24          78           0         102
  6   3   1          10          31           0          41
  6   1   4          10           7           0          17
  6   2   3          46          31           0          77
  6   3   2          46          31           0          77
  6   4   1          10           7           0          17
  6   1   5           3           0           0           3
  6   2   4          24           0           0          24
  6   3   3          46           0           0          46
  6   4   2          24           0           0          24
  6   5   1           3           0           0           3
  6   1   6           1           0           0           1
  6   2   5           3           0           0           3
  6   3   4          10           0           0          10
  6   4   3          10           0           0          10
  6   5   2           3           0           0           3
  6   6   1           1           0           0           1
  6     sum         291         285          48         624
\end{verbatim}
\end{scriptsize}
\newpage
\begin{scriptsize}
\begin{verbatim}
Number of unsensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2       all g
  6   1   1           1           6          13          20
  6   1   2           3          22          13          38
  6   2   1           3          22          13          38
  6   1   3           8          22           0          30
  6   2   2          21          61           0          82
  6   3   1           8          22           0          30
  6   1   4           8           6           0          14
  6   2   3          36          22           0          58
  6   3   2          36          22           0          58
  6   4   1           8           6           0          14
  6   1   5           3           0           0           3
  6   2   4          21           0           0          21
  6   3   3          36           0           0          36
  6   4   2          21           0           0          21
  6   5   1           3           0           0           3
  6   1   6           1           0           0           1
  6   2   5           3           0           0           3
  6   3   4           8           0           0           8
  6   4   3           8           0           0           8
  6   5   2           3           0           0           3
  6   6   1           1           0           0           1
  6     sum         240         211          39         490


Number of rooted hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3       all g
  7   1   1           1          70         469         180         720
  7   1   2          21         560        1183           0        1764
  7   2   1          21         560        1183           0        1764
  7   1   3         105        1050         469           0        1624
  7   2   2         280        2695        1183           0        4158
  7   3   1         105        1050         469           0        1624
  7   1   4         175         560           0           0         735
  7   2   3         889        2695           0           0        3584
  7   3   2         889        2695           0           0        3584
  7   4   1         175         560           0           0         735
  7   1   5         105          70           0           0         175
  7   2   4         889         560           0           0        1449
  7   3   3        1694        1050           0           0        2744
  7   4   2         889         560           0           0        1449
  7   5   1         105          70           0           0         175
  7   1   6          21           0           0           0          21
  7   2   5         280           0           0           0         280
  7   3   4         889           0           0           0         889
  7   4   3         889           0           0           0         889
  7   5   2         280           0           0           0         280
  7   6   1          21           0           0           0          21
  7   1   7           1           0           0           0           1
  7   2   6          21           0           0           0          21
  7   3   5         105           0           0           0         105
  7   4   4         175           0           0           0         175
  7   5   3         105           0           0           0         105
  7   6   2          21           0           0           0          21
  7   7   1           1           0           0           0           1
  7     sum        9152       14805        4956         180       29093

Number of sensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3       all g
  7   1   1           1          10          67          30         108
  7   1   2           3          80         169           0         252
  7   2   1           3          80         169           0         252
  7   1   3          15         150          67           0         232
  7   2   2          40         385         169           0         594
  7   3   1          15         150          67           0         232
  7   1   4          25          80           0           0         105
  7   2   3         127         385           0           0         512
  7   3   2         127         385           0           0         512
  7   4   1          25          80           0           0         105
  7   1   5          15          10           0           0          25
  7   2   4         127          80           0           0         207
  7   3   3         242         150           0           0         392
  7   4   2         127          80           0           0         207
  7   5   1          15          10           0           0          25
  7   1   6           3           0           0           0           3
  7   2   5          40           0           0           0          40
  7   3   4         127           0           0           0         127
  7   4   3         127           0           0           0         127
  7   5   2          40           0           0           0          40
  7   6   1           3           0           0           0           3
  7   1   7           1           0           0           0           1
  7   2   6           3           0           0           0           3
  7   3   5          15           0           0           0          15
  7   4   4          25           0           0           0          25
  7   5   3          15           0           0           0          15
  7   6   2           3           0           0           0           3
  7   7   1           1           0           0           0           1
  7     sum        1310        2115         708          30        4163

Number of unsensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3       all g
  7   1   1           1           8          44          25          78
  7   1   2           3          51         108           0         162
  7   2   1           3          51         108           0         162
  7   1   3          12          91          44           0         147
  7   2   2          33         249         108           0         390
  7   3   1          12          91          44           0         147
  7   1   4          17          51           0           0          68
  7   2   3          90         249           0           0         339
  7   3   2          90         249           0           0         339
  7   4   1          17          51           0           0          68
  7   1   5          12           8           0           0          20
  7   2   4          90          51           0           0         141
  7   3   3         171          91           0           0         262
  7   4   2          90          51           0           0         141
  7   5   1          12           8           0           0          20
  7   1   6           3           0           0           0           3
  7   2   5          33           0           0           0          33
  7   3   4          90           0           0           0          90
  7   4   3          90           0           0           0          90
  7   5   2          33           0           0           0          33
  7   6   1           3           0           0           0           3
  7   1   7           1           0           0           0           1
  7   2   6           3           0           0           0           3
  7   3   5          12           0           0           0          12
  7   4   4          17           0           0           0          17
  7   5   3          12           0           0           0          12
  7   6   2           3           0           0           0           3
  7   7   1           1           0           0           0           1
  7     sum         954        1350         456          25        2785


Number of rooted hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3       all g
  8   1   1           1         126        1869        3044        5040
  8   1   2          28        1470        8526        3044       13068
  8   2   1          28        1470        8526        3044       13068
  8   1   3         196        4410        8526           0       13132
  8   2   2         518       11199       21229           0       32946
  8   3   1         196        4410        8526           0       13132
  8   1   4         490        4410        1869           0        6769
  8   2   3        2436       20684        8526           0       31646
  8   3   2        2436       20684        8526           0       31646
  8   4   1         490        4410        1869           0        6769
  8   1   5         490        1470           0           0        1960
  8   2   4        3985       11199           0           0       15184
  8   3   3        7500       20684           0           0       28184
  8   4   2        3985       11199           0           0       15184
  8   5   1         490        1470           0           0        1960
  8   1   6         196         126           0           0         322
  8   2   5        2436        1470           0           0        3906
  8   3   4        7500        4410           0           0       11910
  8   4   3        7500        4410           0           0       11910
  8   5   2        2436        1470           0           0        3906
  8   6   1         196         126           0           0         322
  8   1   7          28           0           0           0          28
  8   2   6         518           0           0           0         518
  8   3   5        2436           0           0           0        2436
  8   4   4        3985           0           0           0        3985
  8   5   3        2436           0           0           0        2436
  8   6   2         518           0           0           0         518
  8   7   1          28           0           0           0          28
  8   1   8           1           0           0           0           1
  8   2   7          28           0           0           0          28
  8   3   6         196           0           0           0         196
  8   4   5         490           0           0           0         490
  8   5   4         490           0           0           0         490
  8   6   3         196           0           0           0         196
  8   7   2          28           0           0           0          28
  8   8   1           1           0           0           0           1
  8     sum       54912      131307       77992        9132      273343

Number of sensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3       all g
  8   1   1           1          17         237         385         640
  8   1   2           4         187        1072         385        1648
  8   2   1           4         187        1072         385        1648
  8   1   3          26         557        1072           0        1655
  8   2   2          67        1409        2664           0        4140
  8   3   1          26         557        1072           0        1655
  8   1   4          64         557         237           0         858
  8   2   3         309        2597        1072           0        3978
  8   3   2         309        2597        1072           0        3978
  8   4   1          64         557         237           0         858
  8   1   5          64         187           0           0         251
  8   2   4         505        1409           0           0        1914
  8   3   3         946        2597           0           0        3543
  8   4   2         505        1409           0           0        1914
  8   5   1          64         187           0           0         251
  8   1   6          26          17           0           0          43
  8   2   5         309         187           0           0         496
  8   3   4         946         557           0           0        1503
  8   4   3         946         557           0           0        1503
  8   5   2         309         187           0           0         496
  8   6   1          26          17           0           0          43
  8   1   7           4           0           0           0           4
  8   2   6          67           0           0           0          67
  8   3   5         309           0           0           0         309
  8   4   4         505           0           0           0         505
  8   5   3         309           0           0           0         309
  8   6   2          67           0           0           0          67
  8   7   1           4           0           0           0           4
  8   1   8           1           0           0           0           1
  8   2   7           4           0           0           0           4
  8   3   6          26           0           0           0          26
  8   4   5          64           0           0           0          64
  8   5   4          64           0           0           0          64
  8   6   3          26           0           0           0          26
  8   7   2           4           0           0           0           4
  8   8   1           1           0           0           0           1
  8     sum        6975       16533        9807        1155       34470

Number of unsensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3       all g
  8   1   1           1          13         140         226         380
  8   1   2           4         112         596         226         938
  8   2   1           4         112         596         226         938
  8   1   3          19         314         596           0         929
  8   2   2          54         840        1558           0        2452
  8   3   1          19         314         596           0         929
  8   1   4          41         314         140           0         495
  8   2   3         205        1507         596           0        2308
  8   3   2         205        1507         596           0        2308
  8   4   1          41         314         140           0         495
  8   1   5          41         112           0           0         153
  8   2   4         325         840           0           0        1165
  8   3   3         604        1507           0           0        2111
  8   4   2         325         840           0           0        1165
  8   5   1          41         112           0           0         153
  8   1   6          19          13           0           0          32
  8   2   5         205         112           0           0         317
  8   3   4         604         314           0           0         918
  8   4   3         604         314           0           0         918
  8   5   2         205         112           0           0         317
  8   6   1          19          13           0           0          32
  8   1   7           4           0           0           0           4
  8   2   6          54           0           0           0          54
  8   3   5         205           0           0           0         205
  8   4   4         325           0           0           0         325
  8   5   3         205           0           0           0         205
  8   6   2          54           0           0           0          54
  8   7   1           4           0           0           0           4
  8   1   8           1           0           0           0           1
  8   2   7           4           0           0           0           4
  8   3   6          19           0           0           0          19
  8   4   5          41           0           0           0          41
  8   5   4          41           0           0           0          41
  8   6   3          19           0           0           0          19
  8   7   2           4           0           0           0           4
  8   8   1           1           0           0           0           1
  8     sum        4566        9636        5554         678       20434


Number of rooted hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3         g=4       all g
  9   1   1           1         210        5985       26060        8064       40320
  9   1   2          36        3360       42588       63600           0      109584
  9   2   1          36        3360       42588       63600           0      109584
  9   1   3         336       14700       77028       26060           0      118124
  9   2   2         882       37035      189999       63600           0      291516
  9   3   1         336       14700       77028       26060           0      118124
  9   1   4        1176       23520       42588           0           0       67284
  9   2   3        5754      108285      189999           0           0      304038
  9   3   2        5754      108285      189999           0           0      304038
  9   4   1        1176       23520       42588           0           0       67284
  9   1   5        1764       14700        5985           0           0       22449
  9   2   4       13941      108285       42588           0           0      164814
  9   3   3       26004      197896       77028           0           0      300928
  9   4   2       13941      108285       42588           0           0      164814
  9   5   1        1764       14700        5985           0           0       22449
  9   1   6        1176        3360           0           0           0        4536
  9   2   5       13941       37035           0           0           0       50976
  9   3   4       42015      108285           0           0           0      150300
  9   4   3       42015      108285           0           0           0      150300
  9   5   2       13941       37035           0           0           0       50976
  9   6   1        1176        3360           0           0           0        4536
  9   1   7         336         210           0           0           0         546
  9   2   6        5754        3360           0           0           0        9114
  9   3   5       26004       14700           0           0           0       40704
  9   4   4       42015       23520           0           0           0       65535
  9   5   3       26004       14700           0           0           0       40704
  9   6   2        5754        3360           0           0           0        9114
  9   7   1         336         210           0           0           0         546
  9   1   8          36           0           0           0           0          36
  9   2   7         882           0           0           0           0         882
  9   3   6        5754           0           0           0           0        5754
  9   4   5       13941           0           0           0           0       13941
  9   5   4       13941           0           0           0           0       13941
  9   6   3        5754           0           0           0           0        5754
  9   7   2         882           0           0           0           0         882
  9   8   1          36           0           0           0           0          36
  9   1   9           1           0           0           0           0           1
  9   2   8          36           0           0           0           0          36
  9   3   7         336           0           0           0           0         336
  9   4   6        1176           0           0           0           0        1176
  9   5   5        1764           0           0           0           0        1764
  9   6   4        1176           0           0           0           0        1176
  9   7   3         336           0           0           0           0         336
  9   8   2          36           0           0           0           0          36
  9   9   1           1           0           0           0           0           1
  9     sum      339456     1138261     1074564      268980        8064     2829325

Number of sensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3         g=4       all g
  9   1   1           1          24         667        2900         900        4492
  9   1   2           4         374        4736        7070           0       12184
  9   2   1           4         374        4736        7070           0       12184
  9   1   3          38        1634        8560        2900           0       13132
  9   2   2          98        4115       21113        7070           0       32396
  9   3   1          38        1634        8560        2900           0       13132
  9   1   4         132        2616        4736           0           0        7484
  9   2   3         640       12033       21113           0           0       33786
  9   3   2         640       12033       21113           0           0       33786
  9   4   1         132        2616        4736           0           0        7484
  9   1   5         196        1634         667           0           0        2497
  9   2   4        1549       12033        4736           0           0       18318
  9   3   3        2890       21990        8560           0           0       33440
  9   4   2        1549       12033        4736           0           0       18318
  9   5   1         196        1634         667           0           0        2497
  9   1   6         132         374           0           0           0         506
  9   2   5        1549        4115           0           0           0        5664
  9   3   4        4671       12033           0           0           0       16704
  9   4   3        4671       12033           0           0           0       16704
  9   5   2        1549        4115           0           0           0        5664
  9   6   1         132         374           0           0           0         506
  9   1   7          38          24           0           0           0          62
  9   2   6         640         374           0           0           0        1014
  9   3   5        2890        1634           0           0           0        4524
  9   4   4        4671        2616           0           0           0        7287
  9   5   3        2890        1634           0           0           0        4524
  9   6   2         640         374           0           0           0        1014
  9   7   1          38          24           0           0           0          62
  9   1   8           4           0           0           0           0           4
  9   2   7          98           0           0           0           0          98
  9   3   6         640           0           0           0           0         640
  9   4   5        1549           0           0           0           0        1549
  9   5   4        1549           0           0           0           0        1549
  9   6   3         640           0           0           0           0         640
  9   7   2          98           0           0           0           0          98
  9   8   1           4           0           0           0           0           4
  9   1   9           1           0           0           0           0           1
  9   2   8           4           0           0           0           0           4
  9   3   7          38           0           0           0           0          38
  9   4   6         132           0           0           0           0         132
  9   5   5         196           0           0           0           0         196
  9   6   4         132           0           0           0           0         132
  9   7   3          38           0           0           0           0          38
  9   8   2           4           0           0           0           0           4
  9   9   1           1           0           0           0           0           1
  9     sum       37746      126501      119436       29910         900      314493

Number of unsensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3         g=4       all g
  9   1   1           1          17         366        1530         524        2438
  9   1   2           4         213        2500        3759           0        6476
  9   2   1           4         213        2500        3759           0        6476
  9   1   3          27         879        4474        1530           0        6910
  9   2   2          76        2309       11286        3759           0       17430
  9   3   1          27         879        4474        1530           0        6910
  9   1   4          78        1388        2500           0           0        3966
  9   2   3         403        6568       11286           0           0       18257
  9   3   2         403        6568       11286           0           0       18257
  9   4   1          78        1388        2500           0           0        3966
  9   1   5         116         879         366           0           0        1361
  9   2   4         920        6568        2500           0           0        9988
  9   3   3        1743       12067        4474           0           0       18284
  9   4   2         920        6568        2500           0           0        9988
  9   5   1         116         879         366           0           0        1361
  9   1   6          78         213           0           0           0         291
  9   2   5         920        2309           0           0           0        3229
  9   3   4        2747        6568           0           0           0        9315
  9   4   3        2747        6568           0           0           0        9315
  9   5   2         920        2309           0           0           0        3229
  9   6   1          78         213           0           0           0         291
  9   1   7          27          17           0           0           0          44
  9   2   6         403         213           0           0           0         616
  9   3   5        1743         879           0           0           0        2622
  9   4   4        2747        1388           0           0           0        4135
  9   5   3        1743         879           0           0           0        2622
  9   6   2         403         213           0           0           0         616
  9   7   1          27          17           0           0           0          44
  9   1   8           4           0           0           0           0           4
  9   2   7          76           0           0           0           0          76
  9   3   6         403           0           0           0           0         403
  9   4   5         920           0           0           0           0         920
  9   5   4         920           0           0           0           0         920
  9   6   3         403           0           0           0           0         403
  9   7   2          76           0           0           0           0          76
  9   8   1           4           0           0           0           0           4
  9   1   9           1           0           0           0           0           1
  9   2   8           4           0           0           0           0           4
  9   3   7          27           0           0           0           0          27
  9   4   6          78           0           0           0           0          78
  9   5   5         116           0           0           0           0         116
  9   6   4          78           0           0           0           0          78
  9   7   3          27           0           0           0           0          27
  9   8   2           4           0           0           0           0           4
  9   9   1           1           0           0           0           0           1
  9     sum       22641       69169       63378       15867         524      171579


Number of rooted hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3         g=4       all g
 10   1   1           1         330       16401      152900      193248      362880
 10   1   2          45        6930      167013      659340      193248     1026576
 10   2   1          45        6930      167013      659340      193248     1026576
 10   1   3         540       41580      471240      659340           0     1172700
 10   2   2        1410      104115     1154095     1595480           0     2855100
 10   3   1         540       41580      471240      659340           0     1172700
 10   1   4        2520       97020      471240      152900           0      723680
 10   2   3       12180      440440     2068070      659340           0     3180030
 10   3   2       12180      440440     2068070      659340           0     3180030
 10   4   1        2520       97020      471240      152900           0      723680
 10   1   5        5292       97020      167013           0           0      269325
 10   2   4       40935      697250     1154095           0           0     1892280
 10   3   3       75840     1264310     2068070           0           0     3408220
 10   4   2       40935      697250     1154095           0           0     1892280
 10   5   1        5292       97020      167013           0           0      269325
 10   1   6        5292       41580       16401           0           0       63273
 10   2   5       60626      440440      167013           0           0      668079
 10   3   4      179860     1264310      471240           0           0     1915410
 10   4   3      179860     1264310      471240           0           0     1915410
 10   5   2       60626      440440      167013           0           0      668079
 10   6   1        5292       41580       16401           0           0       63273
 10   1   7        2520        6930           0           0           0        9450
 10   2   6       40935      104115           0           0           0      145050
 10   3   5      179860      440440           0           0           0      620300
 10   4   4      288025      697250           0           0           0      985275
 10   5   3      179860      440440           0           0           0      620300
 10   6   2       40935      104115           0           0           0      145050
 10   7   1        2520        6930           0           0           0        9450
 10   1   8         540         330           0           0           0         870
 10   2   7       12180        6930           0           0           0       19110
 10   3   6       75840       41580           0           0           0      117420
 10   4   5      179860       97020           0           0           0      276880
 10   5   4      179860       97020           0           0           0      276880
 10   6   3       75840       41580           0           0           0      117420
 10   7   2       12180        6930           0           0           0       19110
 10   8   1         540         330           0           0           0         870
 10   1   9          45           0           0           0           0          45
 10   2   8        1410           0           0           0           0        1410
 10   3   7       12180           0           0           0           0       12180
 10   4   6       40935           0           0           0           0       40935
 10   5   5       60626           0           0           0           0       60626
 10   6   4       40935           0           0           0           0       40935
 10   7   3       12180           0           0           0           0       12180
 10   8   2        1410           0           0           0           0        1410
 10   9   1          45           0           0           0           0          45
 10   1  10           1           0           0           0           0           1
 10   2   9          45           0           0           0           0          45
 10   3   8         540           0           0           0           0         540
 10   4   7        2520           0           0           0           0        2520
 10   5   6        5292           0           0           0           0        5292
 10   6   5        5292           0           0           0           0        5292
 10   7   4        2520           0           0           0           0        2520
 10   8   3         540           0           0           0           0         540
 10   9   2          45           0           0           0           0          45
 10  10   1           1           0           0           0           0           1
 10     sum     2149888     9713835    13545216     6010220      579744    31998903

Number of sensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3         g=4       all g
 10   1   1           1          34        1649       15308       19344       36336
 10   1   2           5         698       16725       65972       19344      102744
 10   2   1           5         698       16725       65972       19344      102744
 10   1   3          56        4172       47164       65972           0      117364
 10   2   2         144       10434      115478      159608           0      285664
 10   3   1          56        4172       47164       65972           0      117364
 10   1   4         256        9724       47164       15308           0       72452
 10   2   3        1226       44091      206895       65972           0      318184
 10   3   2        1226       44091      206895       65972           0      318184
 10   4   1         256        9724       47164       15308           0       72452
 10   1   5         536        9724       16725           0           0       26985
 10   2   4        4111       69790      115478           0           0      189379
 10   3   3        7606      126519      206895           0           0      341020
 10   4   2        4111       69790      115478           0           0      189379
 10   5   1         536        9724       16725           0           0       26985
 10   1   6         536        4172        1649           0           0        6357
 10   2   5        6081       44091       16725           0           0       66897
 10   3   4       18019      126519       47164           0           0      191702
 10   4   3       18019      126519       47164           0           0      191702
 10   5   2        6081       44091       16725           0           0       66897
 10   6   1         536        4172        1649           0           0        6357
 10   1   7         256         698           0           0           0         954
 10   2   6        4111       10434           0           0           0       14545
 10   3   5       18019       44091           0           0           0       62110
 10   4   4       28852       69790           0           0           0       98642
 10   5   3       18019       44091           0           0           0       62110
 10   6   2        4111       10434           0           0           0       14545
 10   7   1         256         698           0           0           0         954
 10   1   8          56          34           0           0           0          90
 10   2   7        1226         698           0           0           0        1924
 10   3   6        7606        4172           0           0           0       11778
 10   4   5       18019        9724           0           0           0       27743
 10   5   4       18019        9724           0           0           0       27743
 10   6   3        7606        4172           0           0           0       11778
 10   7   2        1226         698           0           0           0        1924
 10   8   1          56          34           0           0           0          90
 10   1   9           5           0           0           0           0           5
 10   2   8         144           0           0           0           0         144
 10   3   7        1226           0           0           0           0        1226
 10   4   6        4111           0           0           0           0        4111
 10   5   5        6081           0           0           0           0        6081
 10   6   4        4111           0           0           0           0        4111
 10   7   3        1226           0           0           0           0        1226
 10   8   2         144           0           0           0           0         144
 10   9   1           5           0           0           0           0           5
 10   1  10           1           0           0           0           0           1
 10   2   9           5           0           0           0           0           5
 10   3   8          56           0           0           0           0          56
 10   4   7         256           0           0           0           0         256
 10   5   6         536           0           0           0           0         536
 10   6   5         536           0           0           0           0         536
 10   7   4         256           0           0           0           0         256
 10   8   3          56           0           0           0           0          56
 10   9   2           5           0           0           0           0           5
 10  10   1           1           0           0           0           0           1
 10     sum      215602      972441     1355400      601364       58032     3202839

Number of unsensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3         g=4       all g
 10   1   1           1          24         883        7866        9970       18744
 10   1   2           5         388        8622       33635        9970       52620
 10   2   1           5         388        8622       33635        9970       52620
 10   1   3          38        2196       24085       33635           0       59954
 10   2   2         110        5676       59772       82472           0      148030
 10   3   1          38        2196       24085       33635           0       59954
 10   1   4         148        5037       24085        7866           0       37136
 10   2   3         746       23303      106787       33635           0      164471
 10   3   2         746       23303      106787       33635           0      164471
 10   4   1         148        5037       24085        7866           0       37136
 10   1   5         298        5037        8622           0           0       13957
 10   2   4        2344       36669       59772           0           0       98785
 10   3   3        4386       66787      106787           0           0      177960
 10   4   2        2344       36669       59772           0           0       98785
 10   5   1         298        5037        8622           0           0       13957
 10   1   6         298        2196         883           0           0        3377
 10   2   5        3391       23303        8622           0           0       35316
 10   3   4       10097       66787       24085           0           0      100969
 10   4   3       10097       66787       24085           0           0      100969
 10   5   2        3391       23303        8622           0           0       35316
 10   6   1         298        2196         883           0           0        3377
 10   1   7         148         388           0           0           0         536
 10   2   6        2344        5676           0           0           0        8020
 10   3   5       10097       23303           0           0           0       33400
 10   4   4       16103       36669           0           0           0       52772
 10   5   3       10097       23303           0           0           0       33400
 10   6   2        2344        5676           0           0           0        8020
 10   7   1         148         388           0           0           0         536
 10   1   8          38          24           0           0           0          62
 10   2   7         746         388           0           0           0        1134
 10   3   6        4386        2196           0           0           0        6582
 10   4   5       10097        5037           0           0           0       15134
 10   5   4       10097        5037           0           0           0       15134
 10   6   3        4386        2196           0           0           0        6582
 10   7   2         746         388           0           0           0        1134
 10   8   1          38          24           0           0           0          62
 10   1   9           5           0           0           0           0           5
 10   2   8         110           0           0           0           0         110
 10   3   7         746           0           0           0           0         746
 10   4   6        2344           0           0           0           0        2344
 10   5   5        3391           0           0           0           0        3391
 10   6   4        2344           0           0           0           0        2344
 10   7   3         746           0           0           0           0         746
 10   8   2         110           0           0           0           0         110
 10   9   1           5           0           0           0           0           5
 10   1  10           1           0           0           0           0           1
 10   2   9           5           0           0           0           0           5
 10   3   8          38           0           0           0           0          38
 10   4   7         148           0           0           0           0         148
 10   5   6         298           0           0           0           0         298
 10   6   5         298           0           0           0           0         298
 10   7   4         148           0           0           0           0         148
 10   8   3          38           0           0           0           0          38
 10   9   2           5           0           0           0           0           5
 10  10   1           1           0           0           0           0           1
 10     sum      121823      513012      698568      307880       29910     1671193


Number of rooted hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3         g=4         g=5       all g
 11   1   1           1         495       39963      696905     2286636      604800     3628800
 11   1   2          55       13200      550011     4606910     5458464           0    10628640
 11   2   1          55       13200      550011     4606910     5458464           0    10628640
 11   1   3         825      103950     2221065     8141100     2286636           0    12753576
 11   2   2        2145      259017     5409019    19571123     5458464           0    30699768
 11   3   1         825      103950     2221065     8141100     2286636           0    12753576
 11   1   4        4950      332640     3465000     4606910           0           0     8409500
 11   2   3       23694     1493525    15014846    19571123           0           0    36103188
 11   3   2       23694     1493525    15014846    19571123           0           0    36103188
 11   4   1        4950      332640     3465000     4606910           0           0     8409500
 11   1   5       13860      485100     2221065      696905           0           0     3416930
 11   2   4      105435     3420835    15014846     4606910           0           0    23148026
 11   3   3      194304     6165478    26717482     8141100           0           0    41218364
 11   4   2      105435     3420835    15014846     4606910           0           0    23148026
 11   5   1       13860      485100     2221065      696905           0           0     3416930
 11   1   6       19404      332640      550011           0           0           0      902055
 11   2   5      216601     3420835     5409019           0           0           0     9046455
 11   3   4      634865     9684433    15014846           0           0           0    25334144
 11   4   3      634865     9684433    15014846           0           0           0    25334144
 11   5   2      216601     3420835     5409019           0           0           0     9046455
 11   6   1       19404      332640      550011           0           0           0      902055
 11   1   7       13860      103950       39963           0           0           0      157773
 11   2   6      216601     1493525      550011           0           0           0     2260137
 11   3   5      931854     6165478     2221065           0           0           0     9318397
 11   4   4     1482250     9684433     3465000           0           0           0    14631683
 11   5   3      931854     6165478     2221065           0           0           0     9318397
 11   6   2      216601     1493525      550011           0           0           0     2260137
 11   7   1       13860      103950       39963           0           0           0      157773
 11   1   8        4950       13200           0           0           0           0       18150
 11   2   7      105435      259017           0           0           0           0      364452
 11   3   6      634865     1493525           0           0           0           0     2128390
 11   4   5     1482250     3420835           0           0           0           0     4903085
 11   5   4     1482250     3420835           0           0           0           0     4903085
 11   6   3      634865     1493525           0           0           0           0     2128390
 11   7   2      105435      259017           0           0           0           0      364452
 11   8   1        4950       13200           0           0           0           0       18150
 11   1   9         825         495           0           0           0           0        1320
 11   2   8       23694       13200           0           0           0           0       36894
 11   3   7      194304      103950           0           0           0           0      298254
 11   4   6      634865      332640           0           0           0           0      967505
 11   5   5      931854      485100           0           0           0           0     1416954
 11   6   4      634865      332640           0           0           0           0      967505
 11   7   3      194304      103950           0           0           0           0      298254
 11   8   2       23694       13200           0           0           0           0       36894
 11   9   1         825         495           0           0           0           0        1320
 11   1  10          55           0           0           0           0           0          55
 11   2   9        2145           0           0           0           0           0        2145
 11   3   8       23694           0           0           0           0           0       23694
 11   4   7      105435           0           0           0           0           0      105435
 11   5   6      216601           0           0           0           0           0      216601
 11   6   5      216601           0           0           0           0           0      216601
 11   7   4      105435           0           0           0           0           0      105435
 11   8   3       23694           0           0           0           0           0       23694
 11   9   2        2145           0           0           0           0           0        2145
 11  10   1          55           0           0           0           0           0          55
 11   1  11           1           0           0           0           0           0           1
 11   2  10          55           0           0           0           0           0          55
 11   3   9         825           0           0           0           0           0         825
 11   4   8        4950           0           0           0           0           0        4950
 11   5   7       13860           0           0           0           0           0       13860
 11   6   6       19404           0           0           0           0           0       19404
 11   7   5       13860           0           0           0           0           0       13860
 11   8   4        4950           0           0           0           0           0        4950
 11   9   3         825           0           0           0           0           0         825
 11  10   2          55           0           0           0           0           0          55
 11  11   1           1           0           0           0           0           0           1
 11     sum    13891584    81968469   160174960   112868844    23235300      604800   392743957

Number of sensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3         g=4         g=5       all g
 11   1   1           1          45        3633       63355      207876       54990      329900
 11   1   2           5        1200       50001      418810      496224           0      966240
 11   2   1           5        1200       50001      418810      496224           0      966240
 11   1   3          75        9450      201915      740100      207876           0     1159416
 11   2   2         195       23547      491729     1779193      496224           0     2790888
 11   3   1          75        9450      201915      740100      207876           0     1159416
 11   1   4         450       30240      315000      418810           0           0      764500
 11   2   3        2154      135775     1364986     1779193           0           0     3282108
 11   3   2        2154      135775     1364986     1779193           0           0     3282108
 11   4   1         450       30240      315000      418810           0           0      764500
 11   1   5        1260       44100      201915       63355           0           0      310630
 11   2   4        9585      310985     1364986      418810           0           0     2104366
 11   3   3       17664      560498     2428862      740100           0           0     3747124
 11   4   2        9585      310985     1364986      418810           0           0     2104366
 11   5   1        1260       44100      201915       63355           0           0      310630
 11   1   6        1764       30240       50001           0           0           0       82005
 11   2   5       19691      310985      491729           0           0           0      822405
 11   3   4       57715      880403     1364986           0           0           0     2303104
 11   4   3       57715      880403     1364986           0           0           0     2303104
 11   5   2       19691      310985      491729           0           0           0      822405
 11   6   1        1764       30240       50001           0           0           0       82005
 11   1   7        1260        9450        3633           0           0           0       14343
 11   2   6       19691      135775       50001           0           0           0      205467
 11   3   5       84714      560498      201915           0           0           0      847127
 11   4   4      134750      880403      315000           0           0           0     1330153
 11   5   3       84714      560498      201915           0           0           0      847127
 11   6   2       19691      135775       50001           0           0           0      205467
 11   7   1        1260        9450        3633           0           0           0       14343
 11   1   8         450        1200           0           0           0           0        1650
 11   2   7        9585       23547           0           0           0           0       33132
 11   3   6       57715      135775           0           0           0           0      193490
 11   4   5      134750      310985           0           0           0           0      445735
 11   5   4      134750      310985           0           0           0           0      445735
 11   6   3       57715      135775           0           0           0           0      193490
 11   7   2        9585       23547           0           0           0           0       33132
 11   8   1         450        1200           0           0           0           0        1650
 11   1   9          75          45           0           0           0           0         120
 11   2   8        2154        1200           0           0           0           0        3354
 11   3   7       17664        9450           0           0           0           0       27114
 11   4   6       57715       30240           0           0           0           0       87955
 11   5   5       84714       44100           0           0           0           0      128814
 11   6   4       57715       30240           0           0           0           0       87955
 11   7   3       17664        9450           0           0           0           0       27114
 11   8   2        2154        1200           0           0           0           0        3354
 11   9   1          75          45           0           0           0           0         120
 11   1  10           5           0           0           0           0           0           5
 11   2   9         195           0           0           0           0           0         195
 11   3   8        2154           0           0           0           0           0        2154
 11   4   7        9585           0           0           0           0           0        9585
 11   5   6       19691           0           0           0           0           0       19691
 11   6   5       19691           0           0           0           0           0       19691
 11   7   4        9585           0           0           0           0           0        9585
 11   8   3        2154           0           0           0           0           0        2154
 11   9   2         195           0           0           0           0           0         195
 11  10   1           5           0           0           0           0           0           5
 11   1  11           1           0           0           0           0           0           1
 11   2  10           5           0           0           0           0           0           5
 11   3   9          75           0           0           0           0           0          75
 11   4   8         450           0           0           0           0           0         450
 11   5   7        1260           0           0           0           0           0        1260
 11   6   6        1764           0           0           0           0           0        1764
 11   7   5        1260           0           0           0           0           0        1260
 11   8   4         450           0           0           0           0           0         450
 11   9   3          75           0           0           0           0           0          75
 11  10   2           5           0           0           0           0           0           5
 11  11   1           1           0           0           0           0           0           1
 11     sum     1262874     7451679    14561360    10260804     2112300       54990    35704007

Number of unsensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3         g=4         g=5       all g
 11   1   1           1          30        1894       32028      104748       28169      166870
 11   1   2           5         650       25442      211149      250674           0      487920
 11   2   1           5         650       25442      211149      250674           0      487920
 11   1   3          50        4890      102033      372579      104748           0      584300
 11   2   2         145       12507      250375      899919      250674           0     1413620
 11   3   1          50        4890      102033      372579      104748           0      584300
 11   1   4         250       15429      158902      211149           0           0      385730
 11   2   3        1272       70364      692895      899919           0           0     1664450
 11   3   2        1272       70364      692895      899919           0           0     1664450
 11   4   1         250       15429      158902      211149           0           0      385730
 11   1   5         680       22439      102033       32028           0           0      157180
 11   2   4        5280      159881      692895      211149           0           0     1069205
 11   3   3        9895      289690     1235766      372579           0           0     1907930
 11   4   2        5280      159881      692895      211149           0           0     1069205
 11   5   1         680       22439      102033       32028           0           0      157180
 11   1   6         932       15429       25442           0           0           0       41803
 11   2   5       10580      159881      250375           0           0           0      420836
 11   3   4       31276      453914      692895           0           0           0     1178085
 11   4   3       31276      453914      692895           0           0           0     1178085
 11   5   2       10580      159881      250375           0           0           0      420836
 11   6   1         932       15429       25442           0           0           0       41803
 11   1   7         680        4890        1894           0           0           0        7464
 11   2   6       10580       70364       25442           0           0           0      106386
 11   3   5       45593      289690      102033           0           0           0      437316
 11   4   4       72417      453914      158902           0           0           0      685233
 11   5   3       45593      289690      102033           0           0           0      437316
 11   6   2       10580       70364       25442           0           0           0      106386
 11   7   1         680        4890        1894           0           0           0        7464
 11   1   8         250         650           0           0           0           0         900
 11   2   7        5280       12507           0           0           0           0       17787
 11   3   6       31276       70364           0           0           0           0      101640
 11   4   5       72417      159881           0           0           0           0      232298
 11   5   4       72417      159881           0           0           0           0      232298
 11   6   3       31276       70364           0           0           0           0      101640
 11   7   2        5280       12507           0           0           0           0       17787
 11   8   1         250         650           0           0           0           0         900
 11   1   9          50          30           0           0           0           0          80
 11   2   8        1272         650           0           0           0           0        1922
 11   3   7        9895        4890           0           0           0           0       14785
 11   4   6       31276       15429           0           0           0           0       46705
 11   5   5       45593       22439           0           0           0           0       68032
 11   6   4       31276       15429           0           0           0           0       46705
 11   7   3        9895        4890           0           0           0           0       14785
 11   8   2        1272         650           0           0           0           0        1922
 11   9   1          50          30           0           0           0           0          80
 11   1  10           5           0           0           0           0           0           5
 11   2   9         145           0           0           0           0           0         145
 11   3   8        1272           0           0           0           0           0        1272
 11   4   7        5280           0           0           0           0           0        5280
 11   5   6       10580           0           0           0           0           0       10580
 11   6   5       10580           0           0           0           0           0       10580
 11   7   4        5280           0           0           0           0           0        5280
 11   8   3        1272           0           0           0           0           0        1272
 11   9   2         145           0           0           0           0           0         145
 11  10   1           5           0           0           0           0           0           5
 11   1  11           1           0           0           0           0           0           1
 11   2  10           5           0           0           0           0           0           5
 11   3   9          50           0           0           0           0           0          50
 11   4   8         250           0           0           0           0           0         250
 11   5   7         680           0           0           0           0           0         680
 11   6   6         932           0           0           0           0           0         932
 11   7   5         680           0           0           0           0           0         680
 11   8   4         250           0           0           0           0           0         250
 11   9   3          50           0           0           0           0           0          50
 11  10   2           5           0           0           0           0           0           5
 11  11   1           1           0           0           0           0           0           1
 11     sum      683307     3843024     7391499     5180472     1066266       28169    18192737


Number of rooted hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3         g=4         g=5       all g
 12   1   1           1         715       88803     2641925    18128396    19056960    39916800
 12   1   2          66       23595     1585584    24656775    75220860    19056960   120543840
 12   2   1          66       23595     1585584    24656775    75220860    19056960   120543840
 12   1   3        1210      235950     8654646    66805310    75220860           0   150917976
 12   2   2        3135      585585    20981337   159762815   178462816           0   359795688
 12   3   1        1210      235950     8654646    66805310    75220860           0   150917976
 12   1   4        9075      990990    19324305    66805310    18128396           0   105258076
 12   2   3       43098     4410120    82897296   280514670    75220860           0   443086044
 12   3   2       43098     4410120    82897296   280514670    75220860           0   443086044
 12   4   1        9075      990990    19324305    66805310    18128396           0   105258076
 12   1   5       32670     1981980    19324305    24656775           0           0    45995730
 12   2   4      245223    13768300   128420004   159762815           0           0   302196342
 12   3   3      449988    24695580   227256510   280514670           0           0   532916748
 12   4   2      245223    13768300   128420004   159762815           0           0   302196342
 12   5   1       32670     1981980    19324305    24656775           0           0    45995730
 12   1   6       60984     1981980     8654646     2641925           0           0    13339535
 12   2   5      666996    19920390    82897296    24656775           0           0   128141457
 12   3   4     1936308    55785870   227256510    66805310           0           0   351783998
 12   4   3     1936308    55785870   227256510    66805310           0           0   351783998
 12   5   2      666996    19920390    82897296    24656775           0           0   128141457
 12   6   1       60984     1981980     8654646     2641925           0           0    13339535
 12   1   7       60984      990990     1585584           0           0           0     2637558
 12   2   6      925190    13768300    20981337           0           0           0    35674827
 12   3   5     3915576    55785870    82897296           0           0           0   142598742
 12   4   4     6195560    87100531   128420004           0           0           0   221716095
 12   5   3     3915576    55785870    82897296           0           0           0   142598742
 12   6   2      925190    13768300    20981337           0           0           0    35674827
 12   7   1       60984      990990     1585584           0           0           0     2637558
 12   1   8       32670      235950       88803           0           0           0      357423
 12   2   7      666996     4410120     1585584           0           0           0     6662700
 12   3   6     3915576    24695580     8654646           0           0           0    37265802
 12   4   5     9032898    55785870    19324305           0           0           0    84143073
 12   5   4     9032898    55785870    19324305           0           0           0    84143073
 12   6   3     3915576    24695580     8654646           0           0           0    37265802
 12   7   2      666996     4410120     1585584           0           0           0     6662700
 12   8   1       32670      235950       88803           0           0           0      357423
 12   1   9        9075       23595           0           0           0           0       32670
 12   2   8      245223      585585           0           0           0           0      830808
 12   3   7     1936308     4410120           0           0           0           0     6346428
 12   4   6     6195560    13768300           0           0           0           0    19963860
 12   5   5     9032898    19920390           0           0           0           0    28953288
 12   6   4     6195560    13768300           0           0           0           0    19963860
 12   7   3     1936308     4410120           0           0           0           0     6346428
 12   8   2      245223      585585           0           0           0           0      830808
 12   9   1        9075       23595           0           0           0           0       32670
 12   1  10        1210         715           0           0           0           0        1925
 12   2   9       43098       23595           0           0           0           0       66693
 12   3   8      449988      235950           0           0           0           0      685938
 12   4   7     1936308      990990           0           0           0           0     2927298
 12   5   6     3915576     1981980           0           0           0           0     5897556
 12   6   5     3915576     1981980           0           0           0           0     5897556
 12   7   4     1936308      990990           0           0           0           0     2927298
 12   8   3      449988      235950           0           0           0           0      685938
 12   9   2       43098       23595           0           0           0           0       66693
 12  10   1        1210         715           0           0           0           0        1925
 12   1  11          66           0           0           0           0           0          66
 12   2  10        3135           0           0           0           0           0        3135
 12   3   9       43098           0           0           0           0           0       43098
 12   4   8      245223           0           0           0           0           0      245223
 12   5   7      666996           0           0           0           0           0      666996
 12   6   6      925190           0           0           0           0           0      925190
 12   7   5      666996           0           0           0           0           0      666996
 12   8   4      245223           0           0           0           0           0      245223
 12   9   3       43098           0           0           0           0           0       43098
 12  10   2        3135           0           0           0           0           0        3135
 12  11   1          66           0           0           0           0           0          66
 12   1  12           1           0           0           0           0           0           1
 12   2  11          66           0           0           0           0           0          66
 12   3  10        1210           0           0           0           0           0        1210
 12   4   9        9075           0           0           0           0           0        9075
 12   5   8       32670           0           0           0           0           0       32670
 12   6   7       60984           0           0           0           0           0       60984
 12   7   6       60984           0           0           0           0           0       60984
 12   8   5       32670           0           0           0           0           0       32670
 12   9   4        9075           0           0           0           0           0        9075
 12  10   3        1210           0           0           0           0           0        1210
 12  11   2          66           0           0           0           0           0          66
 12  12   1           1           0           0           0           0           0           1
 12     sum    91287552   685888171  1805010948  1877530740   684173164    57170880  5201061455

Number of sensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3         g=4         g=5       all g
 12   1   1           1          62        7417      220244     1510846     1588218     3326788
 12   1   2           6        1976      132202     2054974     6268712     1588218    10046088
 12   2   1           6        1976      132202     2054974     6268712     1588218    10046088
 12   1   3         104       19694      721382     5567550     6268712           0    12577442
 12   2   2         265       48846     1748723    13314231    14872428           0    29984493
 12   3   1         104       19694      721382     5567550     6268712           0    12577442
 12   1   4         765       82652     1610617     5567550     1510846           0     8772430
 12   2   3        3605      367645     6908644    23377106     6268712           0    36925712
 12   3   2        3605      367645     6908644    23377106     6268712           0    36925712
 12   4   1         765       82652     1610617     5567550     1510846           0     8772430
 12   1   5        2736      165262     1610617     2054974           0           0     3833589
 12   2   4       20472     1147628    10702449    13314231           0           0    25184780
 12   3   3       37545     2058329    18938994    23377106           0           0    44411974
 12   4   2       20472     1147628    10702449    13314231           0           0    25184780
 12   5   1        2736      165262     1610617     2054974           0           0     3833589
 12   1   6        5102      165262      721382      220244           0           0     1111990
 12   2   5       55633     1660331     6908644     2054974           0           0    10679582
 12   3   4      161455     4649379    18938994     5567550           0           0    29317378
 12   4   3      161455     4649379    18938994     5567550           0           0    29317378
 12   5   2       55633     1660331     6908644     2054974           0           0    10679582
 12   6   1        5102      165262      721382      220244           0           0     1111990
 12   1   7        5102       82652      132202           0           0           0      219956
 12   2   6       77174     1147628     1748723           0           0           0     2973525
 12   3   5      326432     4649379     6908644           0           0           0    11884455
 12   4   4      516507     7259140    10702449           0           0           0    18478096
 12   5   3      326432     4649379     6908644           0           0           0    11884455
 12   6   2       77174     1147628     1748723           0           0           0     2973525
 12   7   1        5102       82652      132202           0           0           0      219956
 12   1   8        2736       19694        7417           0           0           0       29847
 12   2   7       55633      367645      132202           0           0           0      555480
 12   3   6      326432     2058329      721382           0           0           0     3106143
 12   4   5      752940     4649379     1610617           0           0           0     7012936
 12   5   4      752940     4649379     1610617           0           0           0     7012936
 12   6   3      326432     2058329      721382           0           0           0     3106143
 12   7   2       55633      367645      132202           0           0           0      555480
 12   8   1        2736       19694        7417           0           0           0       29847
 12   1   9         765        1976           0           0           0           0        2741
 12   2   8       20472       48846           0           0           0           0       69318
 12   3   7      161455      367645           0           0           0           0      529100
 12   4   6      516507     1147628           0           0           0           0     1664135
 12   5   5      752940     1660331           0           0           0           0     2413271
 12   6   4      516507     1147628           0           0           0           0     1664135
 12   7   3      161455      367645           0           0           0           0      529100
 12   8   2       20472       48846           0           0           0           0       69318
 12   9   1         765        1976           0           0           0           0        2741
 12   1  10         104          62           0           0           0           0         166
 12   2   9        3605        1976           0           0           0           0        5581
 12   3   8       37545       19694           0           0           0           0       57239
 12   4   7      161455       82652           0           0           0           0      244107
 12   5   6      326432      165262           0           0           0           0      491694
 12   6   5      326432      165262           0           0           0           0      491694
 12   7   4      161455       82652           0           0           0           0      244107
 12   8   3       37545       19694           0           0           0           0       57239
 12   9   2        3605        1976           0           0           0           0        5581
 12  10   1         104          62           0           0           0           0         166
 12   1  11           6           0           0           0           0           0           6
 12   2  10         265           0           0           0           0           0         265
 12   3   9        3605           0           0           0           0           0        3605
 12   4   8       20472           0           0           0           0           0       20472
 12   5   7       55633           0           0           0           0           0       55633
 12   6   6       77174           0           0           0           0           0       77174
 12   7   5       55633           0           0           0           0           0       55633
 12   8   4       20472           0           0           0           0           0       20472
 12   9   3        3605           0           0           0           0           0        3605
 12  10   2         265           0           0           0           0           0         265
 12  11   1           6           0           0           0           0           0           6
 12   1  12           1           0           0           0           0           0           1
 12   2  11           6           0           0           0           0           0           6
 12   3  10         104           0           0           0           0           0         104
 12   4   9         765           0           0           0           0           0         765
 12   5   8        2736           0           0           0           0           0        2736
 12   6   7        5102           0           0           0           0           0        5102
 12   7   6        5102           0           0           0           0           0        5102
 12   8   5        2736           0           0           0           0           0        2736
 12   9   4         765           0           0           0           0           0         765
 12  10   3         104           0           0           0           0           0         104
 12  11   2           6           0           0           0           0           0           6
 12  12   1           1           0           0           0           0           0           1
 12     sum     7611156    57167260   150429819   156469887    57017238     4764654   433460014
\end{verbatim}
\end{scriptsize}
\newpage
\begin{scriptsize}
\begin{verbatim}
Number of unsensed hypermaps with d darts, v vertices and e edges.
  d   v   e         g=0         g=1         g=2         g=3         g=4         g=5       all g
 12   1   1           1          41        3836      110914      757977      797345     1670114
 12   1   2           6        1058       66865     1031387     3142703      797345     5039364
 12   2   1           6        1058       66865     1031387     3142703      797345     5039364
 12   1   3          67       10107      362868     2791448     3142703           0     6307193
 12   2   2         195       25594      883711     6689591     7472556           0    15071647
 12   3   1          67       10107      362868     2791448     3142703           0     6307193
 12   1   4         420       41890      808812     2791448      757977           0     4400547
 12   2   3        2086      188410     3481842    11744994     3142703           0    18560035
 12   3   2        2086      188410     3481842    11744994     3142703           0    18560035
 12   4   1         420       41890      808812     2791448      757977           0     4400547
 12   1   5        1443       83460      808812     1031387           0           0     1925102
 12   2   4       11060      583755     5389906     6689591           0           0    12674312
 12   3   3       20565     1050920     9551009    11744994           0           0    22367488
 12   4   2       11060      583755     5389906     6689591           0           0    12674312
 12   5   1        1443       83460      808812     1031387           0           0     1925102
 12   1   6        2651       83460      362868      110914           0           0      559893
 12   2   5       29237      842635     3481842     1031387           0           0     5385101
 12   3   4       85673     2366909     9551009     2791448           0           0    14795039
 12   4   3       85673     2366909     9551009     2791448           0           0    14795039
 12   5   2       29237      842635     3481842     1031387           0           0     5385101
 12   6   1        2651       83460      362868      110914           0           0      559893
 12   1   7        2651       41890       66865           0           0           0      111406
 12   2   6       40348      583755      883711           0           0           0     1507814
 12   3   5      171275     2366909     3481842           0           0           0     6020026
 12   4   4      271482     3696390     5389906           0           0           0     9357778
 12   5   3      171275     2366909     3481842           0           0           0     6020026
 12   6   2       40348      583755      883711           0           0           0     1507814
 12   7   1        2651       41890       66865           0           0           0      111406
 12   1   8        1443       10107        3836           0           0           0       15386
 12   2   7       29237      188410       66865           0           0           0      284512
 12   3   6      171275     1050920      362868           0           0           0     1585063
 12   4   5      394258     2366909      808812           0           0           0     3569979
 12   5   4      394258     2366909      808812           0           0           0     3569979
 12   6   3      171275     1050920      362868           0           0           0     1585063
 12   7   2       29237      188410       66865           0           0           0      284512
 12   8   1        1443       10107        3836           0           0           0       15386
 12   1   9         420        1058           0           0           0           0        1478
 12   2   8       11060       25594           0           0           0           0       36654
 12   3   7       85673      188410           0           0           0           0      274083
 12   4   6      271482      583755           0           0           0           0      855237
 12   5   5      394258      842635           0           0           0           0     1236893
 12   6   4      271482      583755           0           0           0           0      855237
 12   7   3       85673      188410           0           0           0           0      274083
 12   8   2       11060       25594           0           0           0           0       36654
 12   9   1         420        1058           0           0           0           0        1478
 12   1  10          67          41           0           0           0           0         108
 12   2   9        2086        1058           0           0           0           0        3144
 12   3   8       20565       10107           0           0           0           0       30672
 12   4   7       85673       41890           0           0           0           0      127563
 12   5   6      171275       83460           0           0           0           0      254735
 12   6   5      171275       83460           0           0           0           0      254735
 12   7   4       85673       41890           0           0           0           0      127563
 12   8   3       20565       10107           0           0           0           0       30672
 12   9   2        2086        1058           0           0           0           0        3144
 12  10   1          67          41           0           0           0           0         108
 12   1  11           6           0           0           0           0           0           6
 12   2  10         195           0           0           0           0           0         195
 12   3   9        2086           0           0           0           0           0        2086
 12   4   8       11060           0           0           0           0           0       11060
 12   5   7       29237           0           0           0           0           0       29237
 12   6   6       40348           0           0           0           0           0       40348
 12   7   5       29237           0           0           0           0           0       29237
 12   8   4       11060           0           0           0           0           0       11060
 12   9   3        2086           0           0           0           0           0        2086
 12  10   2         195           0           0           0           0           0         195
 12  11   1           6           0           0           0           0           0           6
 12   1  12           1           0           0           0           0           0           1
 12   2  11           6           0           0           0           0           0           6
 12   3  10          67           0           0           0           0           0          67
 12   4   9         420           0           0           0           0           0         420
 12   5   8        1443           0           0           0           0           0        1443
 12   6   7        2651           0           0           0           0           0        2651
 12   7   6        2651           0           0           0           0           0        2651
 12   8   5        1443           0           0           0           0           0        1443
 12   9   4         420           0           0           0           0           0         420
 12  10   3          67           0           0           0           0           0          67
 12  11   2           6           0           0           0           0           0           6
 12  12   1           1           0           0           0           0           0           1
 12     sum     4004055    29107494    75807708    78573507    28602705     2392035   218487504
\end{verbatim}
\end{scriptsize}

\bibliographystyle{jis}

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\bibitem{W1}
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  Combin. Theory Ser. B} {\bf 13} (1972), 192--218.

\bibitem{Wormald81b}
N.~C. Wormald, Counting unrooted planar maps, {\em Discrete Math.} {\bf 36}
  (1981), 205--225.

\bibitem{Wormald81a}
N.~C. Wormald, On the number of planar maps, {\em Canad. J. Math.} {\bf 33}
  (1981), 1--11.

\end{thebibliography}

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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 05C30; Secondary 05C10, 05C65, 05C85.

\noindent \emph{Keywords: } 
map, hypermap, exhaustive generation, isomorphism class.

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\noindent (Concerned with sequences
\seqnum{A006385},
\seqnum{A006387},
\seqnum{A214814},
\seqnum{A214815},
\seqnum{A214816},
\seqnum{A000257},
\seqnum{A118093},
\seqnum{A214817},
\seqnum{A214818},
\seqnum{A003319},
\seqnum{A090371},
\seqnum{A118094},
\seqnum{A214819},
\seqnum{A214820},
\seqnum{A057005},
\seqnum{A214821},
\seqnum{A214822},
\seqnum{A214823},
\seqnum{A215017}, and
\seqnum{A215018}.)



\vspace*{+.1in}
\noindent
Received October 3 2014;
revised versions received   October 6 2014; October 29 2014; February 19 2015; 
February 23 2015.
Published in {\it Journal of Integer Sequences}, May 12 2015.

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