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Date: Thu, 06 Apr 2006 09:30:02 +0200
From: Andrea <frosini@unisi.it>
Subject: Re: Submission to the Journal of Integer Sequences
To: Jeffrey Shallit <shallit@graceland.math.uwaterloo.ca>
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Dear Prof. Shallit,

we have made some changes to the paper according
to referee suggestions (mainly we have added a short abstract),
and according to the latex style guidelines found
in the web page of the Journal.

If any problems will eventually occur, please do not
hesitate to contact me.

Best regards

a.


%%%%%%%%%%%%%%%%%%%%%%%%
Dr. A. Frosini
Universita degli Studi di Siena
Dipartimento di Scienze Matematiche ed Informatiche
Pian dei Mantellini, 44
53100 Siena, Italy
Phone:  +39 577 233705
Fax:    +39 577 233730
e-mail: frosini@unisi.it
%%%%%%%%%%%%%%%%%%%%%%%%

----- Original Message ----- 
From: "Jeffrey Shallit" <shallit@graceland.math.uwaterloo.ca>
To: <frosini@unisi.it>
Cc: <shallit@graceland.math.uwaterloo.ca>
Sent: Tuesday, April 04, 2006 8:53 PM
Subject: Re: Submission to the Journal of Integer Sequences


> Dear Prof. Frosini:
>
> I have now received the referee's report on your paper,
> "On the sequence A079500 and its combinatorial
> interpretations"
> and it is attached below.
>
> As you can see the referee likes your paper and believes it
> should be accepted, but has one minor suggestion.
>
> Please revise your paper in accordance with the referee's
> suggestion and send me a new version.
>
> Please BE SURE your paper conforms to the Journal's latex
> style guidelines, available on the Journal's home page.
> Non-conforming papers CANNOT be published.
>
> Thank you for considering the Journal of Integer Sequences.
>
> Jeffrey Shallit
> Editor-in-Chief
> Journal of Integer Sequences
>
> -----
>
> Referee's report on the paper -
>
> A Frosini and S Rinaldi, On the sequence A079500 and its combinatorial
> interpretations.
>
> The authors show that the sequence A079500 of Sloane enumerates four 
> distinct
> classes of combinatorial objects. They exhibit explicit nontrivial 
> bijections
> between the different classes. They also show that the generating function
> for the sequence is not D-finite.
>
> The results are well presented and interesting and definitely deserving of
> publication in the Journal of Integer Sequences.
>
> My only comment is that an abstract should be added to the paper.
>
> 

--Boundary_(ID_CpbDIwjU034JquXF85welw)
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Content-disposition: attachment; filename=pse_jis.tex

=0D=0A\documentclass[10pt]{article}=0D=0A=0D=0A\usepackage[dvips]{eps=
fig}=0D=0A\usepackage{latexsym}=0D=0A\usepackage{psfig}=0D=0A\usepack=
age{amssymb}=0D=0A\usepackage{algorithm}=0D=0A\usepackage{algorithmic=
}=0D=0A\usepackage{amsfonts}=0D=0A\usepackage{amsmath}=0D=0A\usepacka=
ge{indentfirst}=0D=0A=0D=0A\newtheorem{definition}{Definition}[sectio=
n]=0D=0A\newtheorem{proposition}{Proposition}[section]=0D=0A\newtheor=
em{example}{Example}[section]=0D=0A\newtheorem{theorem}{Theorem}[sect=
ion]=0D=0A\setlength{\parindent}{20pt}=0D=0A\newcommand{\qed}{\hfill\=
square\medskip}=0D=0A=0D=0A\sloppy=0D=0A=0D=0A\begin{document}=0D=
=0A=0D=0A\date{}=0D=0A\newtheorem{definizione}{Definition}=0D=0A\newt=
heorem{osservazione}{Osservazioni}=0D=0A\newtheorem{proprieta}{Propri=
et\`a}=0D=0A\newtheorem{teorema}{Theorem}=0D=0A\newtheorem{proposizio=
ne}{Proposition}=0D=0A\newtheorem{esempio}{Example}=0D=0A\newtheorem{=
lemma}{Lemma}=0D=0A\newcommand{\proof}{\noindent{\bf Proof.\ }}=0D=
=0A\newtheorem{remark}{Remark}=0D=0A\newtheorem{nota}{Note}=0D=0A\new=
theorem{corollario}{Corollary}=0D=0A=0D=0A\author{A. Frosini $^*$=
=0D=0A\and  S. Rinaldi \thanks{Universit\`a di Siena, Dipartimento di=
 Scienze Matematiche e Informatiche, Pian dei=0D=0AMantellini, 44, 53=
100, Siena, Italy, {\tt [frosini, rinaldi]@unisi.it}.}}=0D=0A=0D=0A\t=
itle{On the sequence A079500 and its combinatorial~interpretations}=
=0D=0A=0D=0A\maketitle=0D=0A=0D=0A\begin{abstract}=0D=0AIn this paper=
 we present some combinatorial structures enumerated by the sequence =
A079500 in the Sloane=0D=0AEncyclopedia of Integer Sequences, and det=
ermine simple bijections among these structures. Then we investigate=
=0D=0Athe nature of the generating function of the sequence, and prov=
e that is not differentiably finite.=0D=0A\end{abstract}=0D=0A=0D=
=0A\section{The sequence A079500}=0D=0A=0D=0AThe purpose of the paper=
 is to collect and exploit several=0D=0Acombinatorial properties of t=
he sequence:=0D=0A=0D=0A$$1,1,2,3,5,8,14,24,43, 77, 140, 256, 472,874=
,1628,3045,5719, 10780,20388, \ldots $$=0D=0A=0D=0AThe sequence, whic=
h we will refer to as $(f_n)_{n\geq 0}$=0D=0A(sequence A079500 in \ci=
te{sloane}), is defined by the generating=0D=0Afunction:=0D=0A=0D=
=0A\begin{equation} f(x) =3D \sum _{n\geq 0} f_n \, x^n=3D (1-x) \sum=
 _{i\geq 0} \frac{x^i}{1-2x+x^{i+1}}.=0D=0A\end{equation}=0D=0A=0D=
=0A\noindent In \cite{K} R. Kemp proved that $f_n$ is the number of {=
\em balanced ordered trees} having $n+1$ nodes.=0D=0AThen in \cite{KR=
} A. Knopfmacher and N. Robbins proved that $f_n$ is the number of {\=
em compositions} of the=0D=0Ainteger $n$ for which {\em the largest s=
ummand occurs in the first position}, and that, as $n \to \infty$=
=0D=0A=0D=0A$$ f_n \sim \frac{2^{n}}{n{\mbox{log}2}}(1+\delta(\mbox{l=
og}_2n)), $$=0D=0A=0D=0A\noindent  where $\delta(x)$ is a continuous =
periodic function of period $1$, mean zero, and small amplitude of=
=0D=0Awhich the authors determined the Fourier expansion; in the same=
 paper they proved the nice property that the=0D=0Acoefficient $f_n$ =
is odd if and only if $n=3Dm^2-1$, or $n=3Dm^2$, with $m\geq 1$.=0D=
=0A=0D=0A\medskip=0D=0A=0D=0A=0D=0AIn 2001 Marc Le Brun defined the `=
`numbral  arithmetic" for binary sequences by replacing addition with=
 binary=0D=0Abitwise inclusive-OR, and multiplication by shift-$\&$-O=
R (to the authors' knowledge the only references on=0D=0Anumbral arit=
hmetic can be found on the Sloane database \cite{sloane}. For the bas=
ic definitions, see sequence=0D=0AA048888; for further properties see=
 also sequences A57892, A67139, A67150, A67399, A67398). He conjectur=
ed that=0D=0A$\left ( f_n \right )_{n\geq 1}$ counts the number of di=
visors of the binary expansion of $2^n-1$. The conjecture=0D=0Awas co=
nfirmed by Richard Schroeppel in the same year.=0D=0A=0D=0ALater, the=
 sequence $\left ( f_n \right )_{n\geq 1}$ comes to our attention in =
the context of the enumeration of=0D=0A{\em exact polyominoes}, i.e.,=
 polyominoes that tile the plane by translation \cite{Gi}. In particu=
lar, in=0D=0A\cite{BFRV} it is proved that $f_n$ is the number of {\e=
m pseudo-square parallelogram polyominoes with flat=0D=0Abottom} havi=
ng semi-perimeter equal to $n+1$.=0D=0A=0D=0A\medskip=0D=0A=0D=0AOur =
aim in this paper is to give combinatorial evidence of these facts by=
 showing bijections between the four=0D=0Aclasses enumerated by the s=
equence A079500.=0D=0A=0D=0AIn the last section we study the sequence=
 by an analytical point of view, and investigate the nature of its=
=0D=0Agenerating function $f(x)$, proving that it is not {\em differe=
ntiably finite} (or {\em D-finite}) \cite{stan}.=0D=0A=0D=0AThe gener=
ating functions of the most common solved models in=0D=0Amathematical=
 physics are differentiably finite, and such functions=0D=0Ahave a ra=
ther simple behavior (for instance, the coefficients can=0D=0Abe comp=
uted quickly in a simple way, they have a nice asymptotic=0D=0Aexpans=
ion, they can be handled using computer algebra). On the=0D=0Acontrar=
y, models leading to non D-finite functions are usually=0D=0Aconsider=
ed ``unsolvable" (see \cite{Guttmann,R}).=0D=0A=0D=0ARecently many au=
thors have applied different techniques to prove=0D=0Athe non D-finit=
eness of models arising from physics or statistics=0D=0A\cite{BMP,BMR=
,R}. Guttmann~\cite{Guttmann} developed a numerical=0D=0Amethod for t=
esting the ``solvability" of lattice models based on=0D=0Athe study o=
f the singularities of their {\em anisotropic=0D=0Agenerating functio=
ns}. In many cases the tests allow to state that=0D=0Athe examined mo=
del has not a D-finite generating function=0D=0A\cite{R}.=0D=0A=0D=
=0A=0D=0A=0D=0A\subsection{Four classes enumerated by A079500}=0D=
=0A=0D=0AIn this section we define the four classes of combinatorial=
=0D=0Aobjects that we are going to consider in the paper, and then we=
=0D=0Aprove, using bijective arguments, that they are enumerated by t=
he=0D=0Asequence A079500.=0D=0A=0D=0A\subsubsection{Pseudo-square par=
allelogram polyominoes}=0D=0A=0D=0AIn the plane  $\Bbb Z \times \Bbb =
Z$ a {\em cell} is a unit=0D=0Asquare, and a {\em polyomino} is a fin=
ite connected union of cells=0D=0Ahaving no cut point. Polyominoes ar=
e defined up to translations. A=0D=0A{\em column} ({\em row}) of a po=
lyomino is the intersection=0D=0Abetween the polyomino and an infinit=
e strip of cells whose centers=0D=0Alie on a vertical (horizontal) li=
ne. In a polyomino the {\em=0D=0Aperimeter} is the length of its boun=
dary and the {\em area} is the=0D=0Anumber of its cells. For the main=
 definitions and results=0D=0Aconcerning polyominoes we refer to \cit=
e{stan}.=0D=0A=0D=0AA particular subclass of the class of convex poly=
ominoes consists=0D=0Aof  the {\em parallelogram} polyominoes, define=
d by two lattice=0D=0Apaths that use  north (vertical) and east (hori=
zontal) unitary=0D=0Asteps, and intersect only at their origin and ex=
tremity. These=0D=0Apaths are commonly called the {\em upper} and the=
 {\em lower=0D=0Apath}. Without loss of generality we assume that the=
 upper and=0D=0Alower path of the polyomino start in $(0,0)$. Figure =
\ref{par1}=0D=0Adepicts a parallelogram polyomino having area 14 and=
=0D=0Asemi-perimeter 10. Note that for parallelogram polyominoes the=
=0D=0Asemi-perimeter is equal to the sum of the numbers of its rows a=
nd=0D=0Acolumns.=0D=0A=0D=0A\begin{figure}[htb]=0D=0A\begin{center}=
=0D=0A\epsfig{figure=3Dpar1.eps} \caption{A parallelogram polyomino, =
its=0D=0Aupper and lower paths.} \label{par1}=0D=0A\end{center}=0D=
=0A\end{figure}=0D=0A=0D=0AThe boundary of a parallelogram polyomino =
is conveniently=0D=0Arepresented by a {\em boundary word} defined on =
the alphabet $\{=0D=0A0,1 \}$, where $0$ and $1$ stand for the horizo=
ntal and vertical=0D=0Astep, respectively. The coding follows the bou=
ndary of the=0D=0Apolyomino starting from $(0,0)$ in a clockwise orie=
ntation. For=0D=0Ainstance, the polyomino in Figure~\ref{par1} is rep=
resented by the=0D=0Aword $11011010001011100010.$=0D=0A=0D=0AIf $X=
=3Du_1\ldots u_k$ is a binary word, we indicate by $\overline{X}$ the=
 mirror image of $X$, i.e., the word=0D=0A$u_k\ldots u_1$, and  the l=
ength of $X$ is $|X|=3Dk.$ Moreover $\left | Y \right | _0$, (resp. $=
\left | Y \right |=0D=0A_1$)  indicates the number of occurrences of =
$0$s (resp. $1$s) in $Y$.\\=0D=0A=0D=0ABeauquier and Nivat \cite{Gi} =
introduced the class of {\em pseudo-square} polyominoes, and proved t=
hat each=0D=0Apolyomino of this class may be used to tile the plane b=
y translation. Indeed, let $A$ and $B$ be two discrete=0D=0Apoints on=
 the boundary of a polyomino $P$. Then  $[A,B]$ and $\overline{[A,B]}=
$ denote respectively the paths=0D=0Afrom A to B on  the boundary of =
$P$ traversed in a clockwise and counterclockwise way. The  point $A'=
$ is the=0D=0Aopposite of $A$ on the boundary of $P$ and satisfies $|=
[A,A']|=3D|[A',A]|$. A polyomino $P$ is said to be {\em=0D=0Apseudo-s=
quare} if there are four points A,B,A',B' on its boundary such that $=
B \in [A,A']$,=0D=0A$[A,B]=3D\overline{[B',A']}$, and $[B,A']=3D\over=
line{[A,B']}$ (see Figure \ref{contile}).=0D=0A=0D=0A\begin{figure}[h=
]=0D=0A\begin{center}=0D=0A\epsfig{figure=3Dcontile.eps} \caption{A p=
seudo-square polyomino,=0D=0Aits decomposition and a  tiling of the p=
lane determined by the=0D=0Apolyomino.\label{contile}}=0D=0A\end{cent=
er}=0D=0A\end{figure}=0D=0A=0D=0AHere we consider the class ${\mathca=
l {PSP}}$ of parallelogram=0D=0Apolyominoes which are also pseudo-squ=
ares (briefly, {\em=0D=0Apsp}-polyominoes) \cite{BFRV}.=0D=0A=0D=0A\b=
egin{proposition}\label{prop}=0D=0AIf $X\, Y\, \overline{X}\, \overli=
ne{Y}$ is a decomposition of the=0D=0Aboundary word of a $psp$-polyom=
ino then:=0D=0A=0D=0A\item[i)] $XY$ encodes its upper path, and $YX$ =
its lower path; \item[ii)] $X=3D1X'1$, $Y=3D0Y'0$, for some $X',Y'\in=
=0D=0A\{0,1\}$; \item[iii)] the decomposition $X\, Y\, \overline{X}\,=
 \overline{Y}$ is unique.=0D=0A\end{proposition}=0D=0A=0D=0A\begin{fi=
gure}[htb]=0D=0A\begin{center}=0D=0A\epsfig{figure=3Dqpex.eps} \capti=
on{A {\em psp}-polyomino, and its=0D=0Aunique decomposition.\label{pq=
ex}}=0D=0A\end{center}=0D=0A\end{figure}=0D=0A=0D=0A\noindent For ins=
tance, the polyomino in Figure~\ref{pqex} can be=0D=0Adecomposed as $=
111101\cdot 0100 \cdot 101111 \cdot 0010$, where=0D=0A$X=3D111101$, $=
Y=3D0100$.=0D=0A=0D=0AWe call {\em psp}-polyominoes with {\em flat bo=
ttom} those {\em=0D=0Apsp}-polyominoes such that the word $Y$ (called=
 the {\em bottom})=0D=0Ais made only of $0$'s (see Figure~\ref{planar=
basis}). We denote by=0D=0A${\mathcal {PSP}}^{-}$ the class of such p=
olyominoes, and by=0D=0A${\mathcal {PSP}}^{-}_{n,k}$ those having bot=
tom of length $k\geq=0D=0A1$, and semiperimeter $n+1$. The enumeratio=
n problem for the class=0D=0A${\mathcal {PSP}}^{-}$ is solved in \cit=
e{BFRV}, and here we=0D=0Arecall the useful characterization which le=
ads to such a result:=0D=0A=0D=0A\begin{proposition}\label{propx}=
=0D=0AThe word $U \, =3D \, 1 \, X' \, 1 \, 0^k$, with $k\geq 1$, and=
=0D=0A$|U|=3Dn+1$, represents the upper path of a polyomino in ${\mat=
hcal=0D=0A{PSP}}^{-}_{n,k}$ if and only if $X'$ does not contain any =
factor=0D=0A$0^j$, with $j\geq k$.=0D=0A\end{proposition}=0D=0A=0D=
=0A\begin{example}\label{ex1}=0D=0A{\em The word $1100100011101001100=
01$ represents the upper path of a polyomino in ${\mathcal {PSP}}^{-}=
_{24,4}$,=0D=0Aas shown in Figure~\ref{planarbasis} (a), while the wo=
rd $101100000101$ does not encode a polyomino in ${\mathcal=0D=0A{PSP=
}}^{-}_{15,4}$ since it contains the factor $00000$ (as shown in Figu=
re~\ref{planarbasis} (b)).=0D=0A=0D=0A\begin{figure}[htb]=0D=0A\begin=
{center}=0D=0A\epsfig{figure=3Dplanarbasis.eps,width=3D3.5in,clip=
=3D}\caption{Examples=0D=0Aof a polyomino in ${\mathcal {PSP}}^{-}_{2=
4,4}$=0D=0A (a), and a polygon which is not a polyomino, (b)\label{pl=
anarbasis}}=0D=0A\end{center}=0D=0A\end{figure}=0D=0A}=0D=0A\end{exam=
ple}=0D=0A=0D=0A\subsubsection{Balanced ordered trees}=0D=0A=0D=0AFor=
 the basic definitions of the ordered trees we refer to=0D=0A\cite{st=
an}. An ordered tree is said to be {\em balanced} if all=0D=0Aits lea=
ves are at the same level. Let ${\cal B}_n$ be the class of=0D=0A{\em=
 balanced ordered trees} having $n+1$ nodes. In \cite{K} R.=0D=0AKemp=
 proves that $\left | {\cal B}_n \right | =3D f_n$,~$n\geq 0$.=0D=
=0A=0D=0A\begin{figure}[htb]=0D=0A\begin{center}=0D=0A\epsfig{figure=
=3Dtre_ps.eps,width=3D4.5in,clip=3D} \caption{The eight=0D=0Abalanced=
 ordered trees having $6$ nodes. \label{tre_ps}}=0D=0A\end{center}=
=0D=0A\end{figure}=0D=0A=0D=0A\subsubsection{Compositions with the la=
rgest part in the first position}=0D=0A=0D=0AFor any $n \geq 1$, $k \=
geq 1$ let ${\cal C}_{n,k}$ be the set of compositions of $n$ having =
the largest part $k$=0D=0Ain the first position, i.e.,=0D=0A=0D=0A$$ =
k+a_1+\ldots + a_h=3Dn, $$=0D=0A=0D=0A\noindent with $k\geq a_1, \ldo=
ts ,a_h \geq 1$, and $h\geq 0$. For=0D=0Ainstance:=0D=0A=0D=0A$$ {\ca=
l C}_{5,2} \; =3D \; \bigl\{ \; 2+1+1+1, \; 2+2+1, \; 2+1+2 \; \bigr\=
} .$$=0D=0A=0D=0AMoreover let ${\cal C}_n=3D \bigcup _{k=3D1}^{n} {\c=
al C}_{n,k}$ be=0D=0Athe set of compositions of $n$ having the larges=
t part in the=0D=0Afirst position. For instance:=0D=0A=0D=0A{\small=
=0D=0A=0D=0A$$ {\cal C}_5 =3D \bigl\{ \, 1+1+1+1+1, 2+1+1+1, 2+2+1, 2=
+1+2, 3+2, 3+1+1, 4+1, 5 \, \bigl\}.$$=0D=0A=0D=0A}=0D=0A=0D=0A\noind=
ent By convention we set $\left | {\cal C}_0 \right |=3D\left=0D=0A| =
{\cal C}_1 \right |=3D1$. In \cite{KR} A. Knopfmacher and N.=0D=0ARob=
bins prove that for any $n\geq 0$, $\left | {\cal C}_n \right |=0D=
=0A=3D f_n$.=0D=0A=0D=0A\subsubsection{Divisors of $2^n-1$ in numbral=
 arithmetic}=0D=0A=0D=0ALet $n$ be an integer number, and let $[n]$ b=
e its binary representation. The {\em numbral arithmetic} relies on=
=0D=0Athe replacing of the standard addition for binary sequences wit=
h binary bitwise inclusive-OR. As a consequence,=0D=0Athe multiplicat=
ion uses the shift-$\&$-OR instead of the standard shift-$\&$-add. As=
 an example, it holds=0D=0A$$=0D=0A\begin{array}{lll}=0D=0A&[3] + [9]=
 &=3D 1\,1 + 1\,0\,0\,1 =3D 1\,0\,1\,1 =3D [11] \\ \\=0D=0A&[3] \,* \=
,[9] &=3D 1\,1=0D=0A* 1\,0\,0\,1 =3D 1\,1*1 + 1\,1\,0*0+ 1\,1\,0\,0*0=
+1\,1\,0\,0\,0*1=3D \\ \\=0D=0A& &=3D1\,1+0\,0\,0+0\,0\,0\,0 +1\,1\,0=
\,0\,0=3D 1\,1\,0\,1\,1=3D[27].=0D=0A\end{array}=0D=0A$$=0D=0A=0D=
=0AClearly, the defined addition and multiplication are still=0D=0Aco=
mmutative.=0D=0A=0D=0AWe say that $[d]$ {\em divides} $[n]$ if there =
exists $[e]$ such that $[d] * [e] =3D [n]$. One can apply the given=
=0D=0Adefinitions to compute the six divisors of $[14]$, i.e., $[1]$,=
 $[2]$, $[3]$, $[6]$, $[7]$, and $[14]$, in order=0D=0Ato immediately=
 relying that the element $[e]$ whose product with $[d]$ is $[n]$ is,=
 in general, not unique.=0D=0A=0D=0ALet us define $\mathcal{D}_{n}$ t=
o be the set of the divisors of $[2^n-1]$ (i.e., the binary sequence =
$1^n$), and=0D=0A$\mathcal{D}_{n,k}$ the divisors of $[2^n-1]$ having=
 length $k$. We will show that, for each $n>0$, the=0D=0Acardinality =
of the set $\mathcal{D}_{n}$ is $f_{n}$, by bijectively prove that=
=0D=0A$|\mathcal{D}_{n,k}|=3D|\mathcal{PSP}^-_{n,n-k+1}|$. In the fol=
lowing table the divisors of $[2^n-1]$, for=0D=0A$n=3D1,\dots,4$, are=
 computed.=0D=0A=0D=0A\bigskip=0D=0A=0D=0A\hspace{-1cm}\begin{tabular=
}{c|cccc}=0D=0A{\em n} & $[2^n-1]$ &divisors of $[2^n-1]$&$[\frac{4^n=
-1}{3}]$ & divisors of $[\frac{4^n-1}{3}]$ \\=0D=0A\\=0D=0A  \hline=
=0D=0A\\=0D=0A  1 &1  &\{1\} & 1 & \{1\} \\=0D=0A\\=0D=0A  \hline=
=0D=0A\\=0D=0A  2 &1\, 1 &$\left\{\,1 \,,\, 1\,1 \right\}$&1\, 0\, 1 =
 &$\left\{\,1 \,,\, 1\,0\,1 \right\}$ \\=0D=0A\\=0D=0A  \hline=0D=
=0A\\=0D=0A  3 & 1\, 1\, 1 &\{\, 1 \,,\, 1\,1 \,,\, 1\,1\,1 \} & 1\,0=
\, 1\,0\, 1 & \{\, 1 \,,\,=0D=0A  1\,0\,1 \,,\, 1\, 0\, 1\,0\,1 \} \\=
=0D=0A\\=0D=0A    \hline=0D=0A\\=0D=0A  4 & 1\, 1\, 1\, 1 &$\left\{\b=
egin{array}{c}=0D=0A  1 \,,\, 1\,1 \,,\, 1\,1\,1\\=0D=0A  1\,0\, 1 \,=
,\, 1\,1\,1\,1=0D=0A  \end{array} \right\}$ & 1\,0\, 1\,0\, 1\, 0\,1 =
& $\left\{\begin{array}{c}=0D=0A  1 \,,\, 1\,0\, 1 \,,\, 1\,0\, 1\, 0=
\, 1\\=0D=0A  1\,0\, 0\,0\, 1 \,,\, 1\, 0 \,1\, 0 \, 0\, 1\,0\, 1=
=0D=0A  \end{array} \right\}$\\=0D=0A\end{tabular}=0D=0A=0D=0A\bigski=
p=0D=0A=0D=0AFor the same values of $n$, there are also shown the num=
bers=0D=0A$[\frac{4^n-1}{3}]$ (sequence A002450 in \cite{sloane}) and=
 their=0D=0Acorrespondent divisors. This sequence has two interesting=
=0D=0Acombinatorial interpretations: for $n>0$, it counts both the=
=0D=0Adegree $(n-1)$ numbral power of $5$, and the partial sums of th=
e=0D=0Afirst $(n-1)$ powers of $4$, and it is also strictly connected=
=0D=0Awith the sequence A079500, as stated in the following:=0D=0A=
=0D=0A\begin{proposition}=0D=0AThe binary sequence $x_1\,x_2\,\dots \=
, x_k$ is a divisor of=0D=0A$[2^n-1]$ if and only if the binary seque=
nce $x_1\,0\,x_2\, 0 \,=0D=0A\dots \, 0\, x_k$ is a divisor of $[\fra=
c{4^n-1}{3}]$.=0D=0A\end{proposition}=0D=0A=0D=0A=0D=0A\subsection{Bi=
jective results}=0D=0A=0D=0AAt the very beginning we easily prove tha=
t the number of $psp$-polyominoes having semiperimeter $n+1$ and flat=
=0D=0Abottom, say ${\mathcal {PSP}}^{-}_{n}$, is $f_n$. By Propositio=
n~\ref{propx} it follows that, for any fixed=0D=0A$k\geq 1$, the gene=
rating function of the $psp$-polyominoes having flat bottom of length=
 $k$ is=0D=0A\begin{equation} f_k(x)=3D\frac{x^{k}}{\ 1-x-x^2-x^3-\ld=
ots=0D=0A-x^{k}},\label{ricorda}=0D=0A\end{equation}=0D=0Ahence the g=
enerating function of ${\mathcal {PSP}}^-$ is given by the sum=0D=
=0A\begin{equation} 1 \, + \, \frac{1}{x^2} \sum _{k\geq 1} f_k(x)=
=3D (1-x)=0D=0A\sum_{i\geq 0} \frac{x^i}{1-2x+x^{i+1}}, \label{sw}=
=0D=0A\end{equation}=0D=0A=0D=0A\noindent i.e., the generating functi=
on of A079500.=0D=0A=0D=0A\begin{figure}[htb]=0D=0A\begin{center}=
=0D=0A\epsfig{figure=3Dps.eps,width=3D5in,clip=3D} \caption{The eight=
 {\em=0D=0Apsp}-polyominoes with flat bottom having semi-perimeter eq=
ual to=0D=0A$6$. \label{ps}}=0D=0A\end{center}=0D=0A\end{figure}=0D=
=0A=0D=0A\subsubsection{A bijection between ${\cal B}_n$ and ${\mathc=
al {PSP}}^-_{n}$}=0D=0A=0D=0AWe prove bijectively that $\left | {\cal=
 B}_n \right |=3D \left | {\mathcal {PSP}}^-_{n} \right |$. We do thi=
s by=0D=0Aestablishing a bijection=0D=0A=0D=0A$$ \Theta : {\cal B}_{n=
,k}  \rightarrowtail {\mathcal {PSP}}^{-}_{n,k}.$$=0D=0A=0D=0Awhere $=
{\cal B}_{n,k}$ denotes the set of trees in ${\cal B}_n$ having heigh=
t equal to $k$, thus proving that=0D=0Athis class is counted by $f_k(=
x)$, i.e., the generating function in~(\ref{ricorda}), for any $k\geq=
 1$.=0D=0A=0D=0ALet us start by observing that, for any $k \geq 1$, e=
ach polyomino=0D=0Ain ${\mathcal {PSP}}^{-}_{n,k}$ has all the rows o=
f length $k$.=0D=0ALet $T \in {\cal B}_{n,k}$, and let $e_1, \ldots ,=
e_h$, $h \geq 1$=0D=0Abe the leaves of $T$, from left to right, and f=
or any $i=3D1, \ldots=0D=0A,h-1$ let $n_i$ be the level of node in $T=
$ which is father of the=0D=0Aleaves $e_i$ and $e_{i+1}$, and has min=
imal level (clearly, $0=0D=0A\leq n_i \leq k-1$). Now $\Theta (T)$ is=
 defined as a polyomino=0D=0Awith $h$ rows, each one with exactly $k$=
 cells, and such that for=0D=0Aany $i=3D1, \ldots ,h-1$ the $i+1$ row=
 is placed just above the=0D=0A$i$th row, and moved on the right by $=
k-n_i-1$ cells (see=0D=0AFigure~\ref{trebal}).=0D=0A=0D=0A\medskip=
=0D=0A=0D=0A\begin{figure}[htb]=0D=0A\begin{center}=0D=0A\epsfig{figu=
re=3Dtrebal.eps,width=3D4.5in,clip=3D} \caption{The=0D=0Abijection be=
tween trees in ${\cal B}_{n,k}$, (a), and polyominoes=0D=0Ain ${\math=
cal {PSP}}^{-}_{n,k}$,~(b). \label{trebal}}=0D=0A\end{center}=0D=0A\e=
nd{figure}=0D=0A=0D=0A\medskip=0D=0A=0D=0AThe reader can easily check=
 that $\Theta (T)\in{\mathcal=0D=0A{PSP}}^{-}_{n,k}$, and that $\Thet=
a$ is a bijective function;=0D=0Amoreover from the definition of $\Th=
eta$ it follows  that the=0D=0Anumber of rows of $\Theta (T)$ is equa=
l to the number of leaves of=0D=0A$T$ (see Figure~\ref{balan}).=0D=
=0A=0D=0A\medskip=0D=0A=0D=0A\begin{figure}[htb]=0D=0A\begin{center}=
=0D=0A\epsfig{figure=3Dbalan.eps,width=3D3in,clip=3D} \caption{A bala=
nced tree=0D=0Aof height $3$ and the corresponding polyomino with bot=
tom of=0D=0Alength $3$. \label{balan}}=0D=0A\end{center}=0D=0A\end{fi=
gure}=0D=0A=0D=0A\subsubsection{A bijection between ${\mathcal {PSP}}=
_{n}^-$ and ${\cal C}_n$}=0D=0A=0D=0AIn this paragraph we will descri=
be a bijection between ${\mathcal {PSP}}^{-}_{n,k}$ and ${\cal C}_{n,=
k}$, for any=0D=0A$k\geq 1$=0D=0A=0D=0A$$ \Delta : {\mathcal {PSP}}^{=
-}_{n,k}  \rightarrowtail {\cal C}_{n,k} .$$=0D=0A=0D=0ALet us consid=
er a polyomino $P$ in ${\mathcal {PSP}}^{-}_{n,k}$, and let the word =
encoding its upper path be=0D=0A=0D=0A$$ 1 \; 0^{e_1} \; 1 \; 0^{e_2}=
 \; \ldots \; 1 \; 0^{e_h} \; 1 \; 0^k, $$=0D=0A=0D=0A\noindent with =
$ e_1 + \ldots + e_h + (h+1) +k =3Dn+1$ and, by Proposition~\ref{prop=
x}, $e_i < k$, for $i=3D1, \ldots=0D=0A,h$. Let us define=0D=0A=0D=
=0A$$ \Delta (P)=3D k + ({e_1}+1) + \ldots + ({e_h}+1). $$=0D=0A=0D=
=0AClearly $\Delta (P)$ is a composition of $k + {e_1} + \ldots +=
=0D=0A{e_h} +h=3Dn$ having $k$ as leading summand, thus $ \Delta (P) =
\in=0D=0A{\cal C}_{n,k}$. Moreover, if the polyomino $P$ has $h$ rows=
,=0D=0A$h\geq 1$, then the corresponding composition $\Delta (P)$ has=
 $h$=0D=0Aparts.=0D=0A=0D=0AFor instance, the polyomino in Figure~\re=
f{balan} (with semi-perimeter $13$, bottom of length $3$, and $6$ row=
s)=0D=0Ais represented by the word $1010110011000$, i.e., $10^110^110=
^010^210^010^3, $ and is mapped through $\Delta$ in=0D=0Athe composit=
ion $3+2+2+1+3+1$ of $12$, with greatest summand $3$ and having $6$ p=
arts. It is easy to check that=0D=0A$\Delta$ is a bijection.=0D=0A=
=0D=0A\bigskip=0D=0A=0D=0AThe table below shows the correspondences b=
etween several=0D=0Aparameters in polyominoes of ${\mathcal {PSP}}^-$=
, balanced=0D=0Aordered trees, and compositions with the largest summ=
and in the=0D=0Afirst position.=0D=0A=0D=0A\bigskip=0D=0A=0D=0A=0D=
=0A=0D=0A\hspace{-.9cm}=0D=0A\begin{tabular}{c|c|c|c}=0D=0A&&\\=0D=
=0A  ${\mathcal {PSP}}^-$ &semi-perimeter + 1 &length of the bottom &=
number of rows \\=0D=0A  &&\\=0D=0A  \hline=0D=0A&&\\=0D=0A  ${\cal B=
}$ &number of nodes +1  &height of the tree &number of leaves \\=0D=
=0A&&\\=0D=0A  \hline=0D=0A&&\\=0D=0A  ${\cal C}$ &sum of the terms &=
leading summand &number of summands \\=0D=0A  &&\\=0D=0A\end{tabular}=
=0D=0A=0D=0A\bigskip=0D=0A=0D=0A=0D=0A\subsubsection{A bijection betw=
een ${\cal D}_{n}$ and ${\mathcal {PSP}}_{n}^-$ }=0D=0A=0D=0AWe achie=
ve $|{\cal D}_{n}|=3D|{\mathcal {PSP}}_{n}^-|$ by defining a bijectio=
n $\Omega$ between ${\cal D}_{n,k}$=0D=0Aand ${\mathcal {PSP}}_{n,n-k=
+1}^-$. The following characterization of the elements of ${\cal D}_{=
n}$ is needed.=0D=0A=0D=0A\begin{lemma}\label{lem1}=0D=0AFor each $n,=
d > 0$, it holds that $[d]$ is a divisor of $[2^n-1]$=0D=0Aif and onl=
y if it ends with the digit $1$, and it does not contain=0D=0Aany sub=
sequence of 0s having length greater than $n-k$, with $k$=0D=0Abeing =
the length of $[d]$.=0D=0A\end{lemma}=0D=0A=0D=0A\begin{proof}=0D=
=0A($\Rightarrow$) Let $[d]=3Dd_1\,d_2 \, \dots \, d_k$,=0D=0A$[e]=
=3De_1\,e_2\, \dots \, e_{n-k+1}$, and $[d]\,*\,=0D=0A[e]=3D[2^n-1]=
=3D1^n$ (where the power notation stands for repetition).=0D=0ABy def=
inition, it holds that=0D=0A$$=0D=0A\begin{array}{l}=0D=0A[d]\,*\, [e=
] =3D d_1 \, \dots \, d_{k-1}\, 0 \, *\, e_{n-k+1} \,+\,=0D=0A\dots \=
, + \, d_1 \, \dots \, d_{k-1}\, 0 \, 0^{n-k}\,*\, e_{1} =3D=0D=0Ap_1=
 \, \dots \, p_{n}.=0D=0A\end{array}=0D=0A$$=0D=0AWe proceed by contr=
adiction and we assume that $d_k=3D0$ or that=0D=0Athere exists in $[=
d]$ a sequence of 0s having length $n-k+1$.=0D=0A=0D=0AIn the first c=
ase, it turns out that the digit $p_{n}$ has value 0 since it is the =
sum (i.e., the inclusive-OR)=0D=0Aof $n-k+1$ digits of value 0, and t=
his is not possible.=0D=0A=0D=0AIn the second case, we assume that $[=
d]=3D1\, d_2 \, \dots \,=0D=0Ad_{k-1}\, 1$ contains the sequence of 0=
s $d_h\,\dots \,=0D=0Ad_{h+n-k}$. By definition, the digit $p_{n-h}$ =
of the product=0D=0A$[d]\,*\, [e]$ has value $0$, since it is the sum=
 of $n-k+1$ bits=0D=0Ahaving value 0. Again this is not possible by t=
he hypothesis=0D=0A$[d]\, * \, [e]=3D1^n$.=0D=0A=0D=0A($\Leftarrow$) =
Let us consider the binary sequence=0D=0A$[e]=3D1^{n-k+1}$. Since $[d=
]=3D1\, d_2 \, \dots \, d_{k-1} \, 1$ does=0D=0Anot contain any seque=
nce of 0s having length greater than $n-k$,=0D=0Athen each digit $p_1=
\, \dots \, p_{n}$ of the product=0D=0A$[d]\,*\,[e]$ is the sum of $n=
-k+1$ digits at least one of them=0D=0Ahaving value $1$, so it has va=
lue $1$. The thesis $[d]\, * \,=0D=0A[e]=3D1^n$ is achieved. $\qed$=
=0D=0A\end{proof}=0D=0A=0D=0AWe define the bijection $$\Omega: {\cal =
D}_{n,k}\rightarrowtail=0D=0A{\mathcal {PSP}}_{n,n-k+1}^-$$ as follow=
s: to each divisor $[d]=3D1=0D=0A\, d_2 \, \dots \, d_{k-1} \, 1$ of =
$[2^n-1]$ we associate a=0D=0A$psp$-polyomino whose upper path is rep=
resented by $1 \, d_2 \,=0D=0A\dots \, d_{k-1} \, 1 \, 0^{n-k+1}$.=
=0D=0A=0D=0ABy Lemma~\ref{lem1}, each divisor of $[2^n-1]$ has no sub=
sequences=0D=0Aof 0s of length greater than $n-k$, so, by=0D=0APropos=
ition~\ref{propx}, it encodes an upper path of a=0D=0A$psp$-polyomino=
 having the bottom of length greater than $n-k+1$.=0D=0AThe minimal p=
ossible semi-perimeter for such polyominoes is $n+1$,=0D=0Aas desired=
. On the contrary, the word coding the upper path of a=0D=0Apolyomino=
 in ${\mathcal {PSP}}_{n,n-k+1}^-$ belongs to ${\cal=0D=0AD}_{n,k}$, =
since it is a binary sequence of the type $w_1\,=0D=0A0^{n-k+1}$, wit=
h $w_1$ starting and ending with the digit 1, and=0D=0Acontaining no =
sequences of 0s of length greater than $n-k$.=0D=0A=0D=0AFigure~\ref{=
corr} shows the correspondence between the eight=0D=0Adivisors of $[2=
^5-1]$ and the eight $psp$-polyominoes in=0D=0A${\mathcal {PSP}}_{5}^=
-$.=0D=0A=0D=0A\medskip=0D=0A=0D=0A\begin{figure}[htb]=0D=0A\begin{ce=
nter}=0D=0A\epsfig{figure=3Dps3.eps,width=3D5in,clip=3D} \caption{Eac=
h upper path of an element in ${\mathcal {PSP}}_{5}^-$ is=0D=0Aassoci=
ated with the correspondent divisor of $[2^n-1]$ in ${\mathcal {D}}_{=
5}$. \label{corr}}=0D=0A\end{center}=0D=0A\end{figure}=0D=0A=0D=0A=
=0D=0A%The table below shows the correspondences between several=0D=
=0A%parameters in polyominoes of ${\mathcal {PSP}}^-$, balanced=0D=
=0A%ordered trees, compositions with the largest summand in the first=
=0D=0A%position, and divisors of the sequences $1^n$.=0D=0A%=0D=0A%\b=
igskip=0D=0A%=0D=0A%=0D=0A%=0D=0A%\hspace{-.9cm}=0D=0A%\begin{tabular=
}{c|c|c|c}=0D=0A%  ${\mathcal {PSP}}^-$ &semi-perimeter + 1 &length o=
f the bottom &number of rows \\=0D=0A%  \hline=0D=0A%  ${\cal B}$ &nu=
mber of nodes +1  &height of the tree &number of leaves \\=0D=0A%  \h=
line=0D=0A%  ${\cal C}$ &sum of the terms &leading summand &number of=
 summands \\=0D=0A%  \hline=0D=0A%  ${\cal D}$ &power of 2 -1 &length=
 of a divisor -1&number of 1s in a divisor \\=0D=0A%\end{tabular}=
=0D=0A=0D=0A=0D=0A=0D=0A\subsection{Nature of the generating function=
}=0D=0A=0D=0AFinally, for the sake of completeness, we would like to =
spend a few words on investigating the nature of the=0D=0Agenerating =
function $f(x)$ of the sequence A079500.=0D=0A=0D=0ALet us start by r=
ecalling that a formal power series $u(x)$ with coefficients in $\mat=
hbb C$ is said to be {\em=0D=0Adifferentiably finite} (briefly, {\em =
D-finite}) if it satisfies a (non-trivial) polynomial equation=0D=
=0A=0D=0A$$ q_m(x) u^{(m)}+q_{m-1}(x) u^{(m-1)}+ \ldots +q_1(x) u' +q=
_0(x)u =3D q(x), $$=0D=0A=0D=0A\noindent with $q_0(x), \ldots ,q_m(x)=
 \in {\mathbb C} [x]$, and $q_m(x) \neq 0$ (\cite{stan}, Ch. 6).=0D=
=0A=0D=0AEvery algebraic series is D-finite, while the converse does =
not hold. For example, the generating function $u(x)$=0D=0Aof the seq=
uence ${2n \choose n}^2$ is D-finite, since it satisfies the linear d=
ifferential equation=0D=0A=0D=0A$$ 4u(x)+(32x-1)u'(x)+x(16x-1)u''(x)=
=3D0, $$=0D=0A=0D=0A\noindent while $u(x)$ is not algebraic, as prove=
d for the first=0D=0Atime in \cite{F}.=0D=0A=0D=0A\medskip=0D=0A=0D=
=0AOur aim in this section is to prove that the generating function=
=0D=0Aof the sequence A079500,=0D=0A=0D=0A$$ f(x) =3D (1-x) \sum _{i\=
geq 0} \frac{x^i}{1-2x+x^{i+1}} $$=0D=0A=0D=0A\noindent is not differ=
entiably finite.=0D=0A=0D=0A\medskip=0D=0A=0D=0AIn order to do this w=
e can use the very simple argument, arising=0D=0Afrom the classical t=
heory of linear differential equation, that a=0D=0AD-finite power ser=
ies of a single variable has only a finite=0D=0Anumber of singulariti=
es. Thus we can reach our goal by proving=0D=0Athat $f(x)$ has infini=
tely many poles. For instance, the function=0D=0A$1 / \cos (x)$ is no=
t D-finite, since it has an infinite number of=0D=0Asingularities.=
=0D=0A=0D=0A\medskip=0D=0A=0D=0AWe point out that this ``criterion" w=
as applied by Flajolet in \cite{F} to prove that the language=0D=0A=
=0D=0A$$ \left \{ \, a^n b v_1 a^n v_2 \, : \, n\geq 1, v_1, v_2 \in =
\{ a,b \}^* \, \right \} $$=0D=0A=0D=0A\noindent is a context-free in=
herently ambiguous language.=0D=0A=0D=0A\medskip=0D=0A=0D=0AWe reach =
our goal by adapting Flajolet's proof to our case. For each $i\geq 1$=
, let $P_i=3D1-2x+x^{i+1}$. For=0D=0A$|x|<1$, it is easy to check tha=
t each $P_i$ has a real zero $\rho _i$ such that $\frac{1}{2} < \rho_=
i < 1$. By=0D=0Athe Principle of the Argument and by the related Rouc=
h\'e's Theorem we are able to state that for a sufficiently=0D=0Asmal=
l $x$, say $|x|<\frac{3}{4}$, $\rho_i$ is the unique zero of $P_i$. T=
hus we have:=0D=0A=0D=0A\begin{enumerate}=0D=0A\item if $i>j$ then $\=
rho_i < \rho_j$; \item $\lim _{i\to \infty}  \rho_i =3D \frac{1}{2}$.=
=0D=0A\end{enumerate}=0D=0A=0D=0AHence, for any $x\neq \rho_i, \frac{=
1}{2}$, and $|x|<\frac{3}{4}$ the series $\sum _{i \geq 0} \frac{1}{P=
_i}$ is=0D=0Aconvergent. As a conclusion, $f(x)$ is analytic in $|x|<=
\frac{3}{4}$ except for finitely many poles $\rho_i$=0D=0Awhich accum=
ulate in $\frac{1}{2}$, thus it is not D-finite.=0D=0A=0D=0AIt is int=
eresting to observe that, while parallelogram polyominoes are one of =
easiest (and most frequently=0D=0Atreated in literature) classes of p=
olyominoes, and have an algebraic generating function according to th=
e=0D=0Asemi-perimeter, {\em psp}-polyominoes with a flat bottom (whic=
h are a rather simple type of parallelogram=0D=0Apolyominoes) are not=
 {\em D-solvable}.=0D=0A=0D=0A\medskip=0D=0A=0D=0A\paragraph{On the n=
ature of a class of languages.} The result=0D=0Ain the present sectio=
n suggests some further considerations=0D=0Aregarding the class ${\ca=
l PSP^-}$ of {\em psp}-polyominoes with=0D=0Aflat bottom.=0D=0A=0D=
=0ABy Proposition~\ref{propx} we have that the class of the polyomino=
es having bottom of length $k\geq 1$ can be=0D=0Asuitably encoded by =
means of the regular language $\ell _k$ defined by the unambiguous re=
gular expression=0D=0A=0D=0A$$ 1 \, \bigl( \, 1 \cup 01 \cup 001 \cup=
 \ldots \cup 0^{k-1}1 \, \bigr)^* \, 0^k, $$=0D=0A=0D=0Aand that cons=
equently the class ${\cal PSP^-}$ can be encoded by=0D=0Ameans of the=
 language $\ell =3D \bigcup_{k\geq 1} \ell _k$.=0D=0A=0D=0AA classica=
l result by Chomsky and Sch\"uetzenberger~\cite{CS} states that the g=
enerating function of an=0D=0Aunambiguous context-free language is al=
gebraic. But, as we have proved in the present section, the generatin=
g=0D=0Afunction of $\ell$ (i.e., $x^2 f(x)$) is not algebraic (actual=
ly, not even $D$-finite), thus $\ell$ is an=0D=0A{inherently ambiguou=
s} language.=0D=0A=0D=0A\begin{thebibliography}{13}=0D=0A=0D=0A\bibit=
em[BN]{Gi}=0D=0AD. Beauquier, M. Nivat, On Translating one Polyomino =
to Tile the Plane, {\it Discrete Comput. Geom. } {\bf 6}=0D=0A(1991) =
575--592.=0D=0A=0D=0A\bibitem[BMP]{BMP}=0D=0AM. Bousquet-M\'elou, Mar=
ko Petkovsek, \newblock{\rm Walks confined in a quadrant are not alwa=
ys D-finite},=0D=0A\newblock{\it Theoret. Comput. Sci.} {\bf 307} (20=
03) 257-276.=0D=0A=0D=0A\bibitem[BMR]{BMR}=0D=0AM. Bousquet-M\'elou, =
A. Rechnitzer, \newblock{\rm The site-perimeter of bargraphs}, \newbl=
ock{\it Advances in=0D=0AApplied Mathematics}, {\bf 31} (2003) 86-112=
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r =0D=0A% Polyline=0D=0An 6450 4500 m 6450 4200 l 6750 4200 l 6750 40=
50 l 6900 4050 l 6900 3900 l=0D=0A 6750 3900 l 6750 3600 l 6600 3600 =
l 6600 3750 l 6450 3750 l=0D=0A 6450 3900 l 6150 3900 l 6150 4200 l 6=
300 4200 l 6300 4500 l=0D=0A=0D=0A 6450 4500 l  cp gs col0 s gr =0D=
=0A% Polyline=0D=0An 6150 4200 m 6150 3900 l 6450 3900 l 6450 3750 l =
6600 3750 l 6600 3600 l=0D=0A 6450 3600 l 6450 3300 l 6300 3300 l 630=
0 3450 l 6150 3450 l=0D=0A 6150 3600 l 5850 3600 l 5850 3900 l 6000 3=
900 l 6000 4200 l=0D=0A=0D=0A 6150 4200 l  cp gs col12 1.00 shd ef gr=
 gs col0 s gr =0D=0A% Polyline=0D=0An 6600 3600 m 6600 3300 l 6900 33=
00 l 6900 3150 l 7050 3150 l 7050 3000 l=0D=0A 6900 3000 l 6900 2700 =
l 6750 2700 l 6750 2850 l 6600 2850 l=0D=0A 6600 3000 l 6300 3000 l 6=
300 3300 l 6450 3300 l 6450 3600 l=0D=0A=0D=0A 6600 3600 l  cp gs col=
0 s gr =0D=0A% Polyline=0D=0An 6300 3300 m 6300 3000 l 6600 3000 l 66=
00 2850 l 6750 2850 l 6750 2700 l=0D=0A 6600 2700 l 6600 2400 l 6450 =
2400 l 6450 2550 l 6300 2550 l=0D=0A 6300 2700 l 6000 2700 l 6000 300=
0 l 6150 3000 l 6150 3300 l=0D=0A=0D=0A 6300 3300 l  cp gs col0 s gr =
=0D=0A% Polyline=0D=0An 6000 5100 m 6000 4800 l 6300 4800 l 6300 4650=
 l 6450 4650 l 6450 4500 l=0D=0A 6300 4500 l 6300 4200 l 6150 4200 l =
6150 4350 l 6000 4350 l=0D=0A 6000 4500 l 5700 4500 l 5700 4800 l 585=
0 4800 l 5850 5100 l=0D=0A=0D=0A 6000 5100 l  cp gs col0 s gr =0D=
=0A% Polyline=0D=0An 5400 4500 m 5400 4200 l 5700 4200 l 5700 4050 l =
5850 4050 l 5850 3900 l=0D=0A 5700 3900 l 5700 3600 l 5550 3600 l 555=
0 3750 l 5400 3750 l=0D=0A 5400 3900 l 5100 3900 l 5100 4200 l 5250 4=
200 l 5250 4500 l=0D=0A=0D=0A 5400 4500 l  cp gs col0 s gr =0D=0A% El=
lipse=0D=0A7.500 slw=0D=0An 1200 4200 54 54 0 360 DrawEllipse gs 0.00=
 setgray ef gr gs col0 s gr=0D=0A=0D=0A% Ellipse=0D=0An 2700 3600 54 =
54 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr=0D=0A=0D=0A% =
Ellipse=0D=0An 2100 3000 54 54 0 360 DrawEllipse gs 0.00 setgray ef g=
r gs col0 s gr=0D=0A=0D=0A% Ellipse=0D=0An 1800 4800 54 54 0 360 Draw=
Ellipse gs 0.00 setgray ef gr gs col0 s gr=0D=0A=0D=0A% Polyline=0D=
=0An 1200 3900 m 1500 3900 l 1500 4200 l 1200 4200 l=0D=0A 1200 3900 =
l  cp gs col0 s gr =0D=0A% Polyline=0D=0An 1200 3600 m 1500 3600 l 15=
00 3900 l 1200 3900 l=0D=0A 1200 3600 l  cp gs col0 s gr =0D=0A% Poly=
line=0D=0An 1500 3600 m 1800 3600 l 1800 3900 l 1500 3900 l=0D=0A 150=
0 3600 l  cp gs col0 s gr =0D=0A% Polyline=0D=0An 1500 3900 m 1800 39=
00 l 1800 4200 l 1500 4200 l=0D=0A 1500 3900 l  cp gs col0 s gr =0D=
=0A% Polyline=0D=0An 1500 4200 m 1800 4200 l 1800 4500 l 1500 4500 l=
=0D=0A 1500 4200 l  cp gs col0 s gr =0D=0A% Polyline=0D=0An 1800 3900=
 m 2100 3900 l 2100 4200 l 1800 4200 l=0D=0A 1800 3900 l  cp gs col0 =
s gr =0D=0A% Polyline=0D=0An 1800 3600 m 2100 3600 l 2100 3900 l 1800=
 3900 l=0D=0A 1800 3600 l  cp gs col0 s gr =0D=0A% Polyline=0D=0An 21=
00 3600 m 2400 3600 l 2400 3900 l 2100 3900 l=0D=0A 2100 3600 l  cp g=
s col0 s gr =0D=0A% Polyline=0D=0An 2100 3300 m 2400 3300 l 2400 3600=
 l 2100 3600 l=0D=0A 2100 3300 l  cp gs col0 s gr =0D=0A% Polyline=
=0D=0A30.000 slw=0D=0An 1200 4200 m 1200 3600 l 1800 3600 l 1800 3300=
 l 2100 3300 l 2100 3000 l=0D=0A 2400 3000 l 2400 3600 l 2700 3600 l =
2700 3900 l 2400 3900 l=0D=0A 2400 4200 l 1800 4200 l 1800 4800 l 150=
0 4800 l 1500 4200 l=0D=0A=0D=0A 1200 4200 l  cp gs col0 s gr =0D=
=0A/Times-BoldItalic ff 270.00 scf sf=0D=0A825 4350 m=0D=0Ags 1 -1 sc=
 (A) col0 sh gr=0D=0A/Times-BoldItalic ff 270.00 scf sf=0D=0A1800 292=
5 m=0D=0Ags 1 -1 sc (B) col0 sh gr=0D=0A/Times-BoldItalic ff 270.00 s=
cf sf=0D=0A1950 5025 m=0D=0Ags 1 -1 sc (B') col0 sh gr=0D=0A% Polylin=
e=0D=0A15.000 slw=0D=0An 5700 4800 m 5700 4500 l 6000 4500 l 6000 435=
0 l 6150 4350 l 6150 4200 l=0D=0A 6000 4200 l 6000 3900 l 5850 3900 l=
 5850 4050 l 5700 4050 l=0D=0A 5700 4200 l 5400 4200 l 5400 4500 l 55=
50 4500 l 5550 4800 l=0D=0A=0D=0A 5700 4800 l  cp gs col0 s gr =0D=
=0A% here ends figure;=0D=0A$F2psEnd=0D=0Ars=0D=0Ashowpage=0D=0A=

--Boundary_(ID_CpbDIwjU034JquXF85welw)--

