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\begin{center}
\vskip 1cm{\LARGE\bf 
Counting Young Tableaux of Bounded Height
}
\vskip 1cm
\large
Fran\c cois Bergeron and Francis Gascon\footnote{
With support from NSERC and FCAR} \\
D\'epartement de Mathematiques \\
Univerit\'e du Quebec \` a Montreal \\
\href{mailto:bergeron.francois@uqam.ca}{\tt bergeron.francois@uqam.ca} \\
\href{mailto:gascon.francis@uqam.ca}{\tt gascon.francis@uqam.ca} \\
\end{center}

\vskip .2 in
\begin{abstract}
We show that formulae of Gessel for the generating functions
for Young standard tableaux of height bounded by $k$
satisfy linear differential equations, with polynomial
coefficients, equivalent to $P$-recurrences conjectured by
Favreau, Krob
and the first author for the number of bounded height
tableaux and pairs of bounded height tableaux.
\end{abstract}

\def\N{{\Bbb N}}
\def\Z{{\Bbb Z}}
\def\R{{\Bbb R}}
\def\C{{\Bbb C}}
\def\Q{{\Bbb Q}}
\def\Tau{{T}}


\def\frac#1#2{{{#1}\over{#2}}}
\def\floor#1{{\left\lfloor #1 \right\rfloor}}

\section{ Results}
Let us first fix some notation. A partition $\lambda$ of a positive
integer $n$ is
a sequence of integers
  $$\lambda_1\geq \lambda_2\geq \ldots\geq \lambda_k&gt;0$$
such that $\sum_i\lambda_i=n$. We denote this by writing $\lambda\vdash n$,
and
say that $k$ is the {\sl height\/} $h(\lambda)$ of $\lambda$. The
height of the empty partition (of 0) is 0.
The (Ferrer's) diagram of a partition $\lambda$ is the set of points
$(i,j)\in\Z^2$ such that
$1\leq j\leq\lambda_i$. It is also denoted by $\lambda$. Clearly a
partition is characterized by its diagram. The conjugate $\lambda'$
of a partition $\lambda$  is the partition with diagram $\{(j,i)\ |\
(i,j)\in\lambda\}$.


A standard Young tableau $T$ is an injective labeling of a Ferrer's
diagram by the elements of
$\{1,2,\ldots,n\}$ such that $T(i,j)&lt;T(i+1,j)$ for $1\leq i&lt; k$ and
$T(i,j)&lt;T(i,j+1)$ for $1\leq j&lt;\lambda_i$.
We further say that $\lambda$ is the {\sl shape\/} of the tableau $T$.
For a given $\lambda$, the number $f_\lambda$ of tableaux of shape
$\lambda$ is given by the {\sl hook length\/} formula
  $$f_\lambda=\frac{n!}{\prod_c h_c},$$
where $c=(i,j)$ runs over the set of points in the diagram of $\lambda$,
and
 $$h_c=\lambda_i-i+\lambda_j'-j+1.$$
Other classical results in this context are
  $$\sum_{\lambda\vdash n}f_\lambda^2 = n!,$$
and
  $$\sum_{\lambda\vdash n}f_\lambda = \ {\hbox{coeff of}} \
\frac{x^n}{n!}\ {\hbox{in}}\ e^{x+x^2/2}.$$

We are interested in the enumeration of tableaux of height bounded by
some integer $k$; that is to say
we wish to compute the numbers
  $$\tau_k(n)=\sum_{h(\lambda)\leq k}f_\lambda,\eqno{(1)}$$
as well as
  $$\Tau_k(n)=\sum_{h(\lambda)\leq k}f_\lambda^2.\eqno{(2)}$$
For example, the first few sequences $\tau_k(n)$ for $n\geq 1$ are

$\tau_2(n)\rightarrow\,1,\,2,\,3,\,6,\,10,\,20,\,35,\,70,\,126,\,252,\,462,\,924,\,1716,\,3432,\,6435,\,12870,\, 24310,\,48620,\, \ldots$

$
\tau_3(n)\rightarrow\,1,\,2,\,4,\,9,\,21,\,51,\,127,\,323,\,835,\,2188,\,5798,\,15511,\,41835,\,
113634,\,310572,\,853467,\ \ldots$

$
\tau_4(n)\rightarrow\,1,\,2,\,4,\,10,\,25,\,70,\,196,\,588,\,1764,\,5544,\,17424,\,56628,\,184041,\,
613470,\,2044900,\ \ldots $

$
\tau_5(n)\rightarrow\,1,\,2,\,4,\,10,\,26,\,75,\,225,\,715,\,2347,\,7990,\,27908,\,99991,\,365587,\,
1362310,\,5159208,\ \ldots $

$
\tau_6(n)\rightarrow\,1,\,2,\,4,\,10,\,26,\,76,\,231,\,756,\,2556,\,9096,\,33231,\,126060,\,488488,\,
1948232,\,7907185,\ \ldots $

\noindent
(These are sequences
\seqnum{A001405},
\seqnum{A001006},
\seqnum{A005817},
\seqnum{A049401}, and
\seqnum{A007579}
in \cite{eis}.)
For $\Tau_k(n)$, we have

$\Tau_2(n)\rightarrow\,1,\,2,\,5,\,14,\,42,\,132,\,429,\,1430,\,4862,\,
16796,\,58786,\,208012,\, 742900,\,2674440,\,9694845,\ \ldots$

$\Tau_3(n)\rightarrow\,1,\,2,\,6,\,23,\,103,\,513,\,2761,\,15767,\,
94359,\,586590,\,3763290,\, 24792705,\,167078577,\ \ldots$

$\Tau_4(n)\rightarrow\,1,\,2,\,6,\,24,\,119,\,694,\,4582,\,33324,\,
261808,\,2190688,\,19318688,\, 178108704,\,1705985883,\ \ldots$

$\Tau_5(n)\rightarrow\,1,\,2,\,6,\,24,\,120,\,719,\,5003,\,39429,\,
344837,\,3291590,\,33835114,\, 370531683,\,4285711539,\ \ldots$

$\Tau_6(n)\rightarrow\,1,\,2,\,6,\,24,\,120,\,720,\,5039,\,40270,\,
361302,\,3587916,\,38957991,\, 457647966,\,5763075506,\ \ldots$

\noindent
(Sequences
\seqnum{A000108},
\seqnum{A005802},
\seqnum{A052397},
\seqnum{A052398}, and
\seqnum{A052399}
in \cite{eis}.)


In \cite{G} Gessel  deduces
the following formulae
from a result of Gordon:
  $$ y_k(x):=\sum _{n=0}^{\infty }{\frac
{\tau_{{k}}(n){x}^{n}}{n!}}= \begin{cases}
\det
\left[J_{i-j}(x)-J_{i+j-1}(x)\right]_{1\leq i,j\leq k/2}&amp;, \text{if $k$ is
even,} \\
e^x\,\det \left[J_{i-j}(x)-J_{i+j}(x)\right]_{1\leq
i,j\leq (k-1)/2}, &amp; \text{if $k$ is odd};  
\end{cases}$$  
and
  $$ Y_k(x):=\sum _{n=0}^{\infty }{\frac {\Tau_{{k}}(n){x}^{n}}{
   \left (n!\right )^{2}}}=\det \left[I_{i-j}(x)\right]_{1\leq i,j\leq
k}, $$
where
$$ J_k(x)=\sum _{n=0}^{\infty }{\frac {{x}^{2\,n+k}}{n!\,\left (n+k
\right )!}} $$
and
$$ I_k(x)=\sum _{n=0}^{\infty }{\frac {{x}^{n+k/2}}{n!\,\left (n+k
\right )!}} $$
If $k$ is positive integer, we set $J_{-k}:=J_k$ and $I_{-k}:=I_k$. The
resulting expressions rapidly become unwieldy. For example,

$ y_2(x)=J_{0}(x)+J_1(x) $

$ y_3(x)={e^{x}}\left(J_{0}(x)-J_2(x)\right) $

$  y_4(x)=J_{0}(x)^2+J_{0}(x)\,J_1(x)+J_{0}(x)\,J_3(x)-J_1(
x)^2-2\,J_1(x)\,J_2(x)+J_1(x)\,J_3(x)-J_2(x)^2 $

$ y_5(x)=e^{x}\left( J_{0}(x)^2- J_{0}(x)\,J_2(x)
- J_{0}(x)\,J_4(x)- J_1(x)^2+2\, J_1(x)
\,J_3(x)+  J_2(x)\,J_4(x)- J_3(x)^2
\right) $

\begin{eqnarray*} y_6(x)&amp;= &amp; J_{0}(x)^3+J_{0}(x)^2\,J_1(x)+J_{0}(x)^2\,J_3(x)+J_{0}(x)^2\,J_5(x)-2\,J_{0}(x)\,J_1(x)^2 -2\,J_{0}(x)\,J_1(x)\,J_2(x) \\
&amp;\qquad &amp; +J_{0}(x)\,J_1(x)\,J_3(x)-2\,J_{0}(x)\,J_1(x)\,J_4(x)+J_{0}(x)\,J_1(x)\, J_5(x)-2\,J_{0}(x)\,J_2(x)^2 \\
&amp;\qquad &amp; -2\,J_{0}(x)\,J_2(x)\,J_3(x)-J_{0}(x)\,J_3(x)^2+J_{0}(x)\,J_3(x)\,J_5(x)-J_{0}(x)\,J_4(x)^2-J_1(x)^3 \\
&amp;\qquad &amp; +2\,J_1(x)^2\,J_2(x)+2\,J_1(x)^2\,J_3(x)-2\,J_1(x)^2\,J_4(x)-J_1(x)^2\,J_5(x)+2\,J_1(x)\,J_2(x)^2 \\
&amp;\qquad &amp; +2\,J_1(x)\,J_2(x)\,J_3(x)+2\,J_1(x)\,J_2(x)\,J_4(x)-2\,J_1(x)\,J_2(x)\,J_5(x)+2\,J_1(x)\,J_3(x)\,J_4(x) \\
&amp;\qquad &amp; +J_1(x)\,J_3(x)\,J_5(x)- J_1(x)\,J_4(x)^2-J_2(x)^2\,J_3(x)+2\,J_2(x)^2\,J_4(x)-J_2(x)^2\,J_5(x) \\
&amp;\qquad &amp; -2\,J_2(x)\,J_3(x)^2+2\,J_2(x)\,J_3(x)\,J_4(x)-J_3(x)^3.  
\end{eqnarray*}

We can simplify these
using properties of Bessel functions.
Recalling the easily deduced relations
   $$ J_{k}(x)=J_{k-2}(x)-{1\over x}(n-k-1)\,J_{k-1}(x), \qquad k\geq
2,$$
we get, after some computation, the much simpler expressions

$ y_3(x)={x^{-1}{e^{x}}}\,J_1(x)$

$
y_4(x)={x^{-2}}\,\left(-2\,x\,J_{0}(x)^2+2\,J_{0}(x)\,J_1(x)+(2\,x+1)\,J_1(x)^2\right)$

$ y_5(x)={
x^{-4}{e^{x}}}\left(-4\,\,x^2\,J_{0}(x)^2+2\,x\,J_{0}(x)\,J_1(x)
+2\,\left(2\,x^2+1\right)J_1(x)^2\right) $

\begin{eqnarray*}
y_6(x) &amp;=&amp; {x^{-6}}\left(
-4\,x^2\,(4\,x-3)\,J_{0}(x)^3
-4\,x\,\left(4\,x^2-3\,x+6\right)J_{0}(x)^2\,J_1(x)
\right. \\
&amp;\qquad&amp; \left.{}+4\,
\left(4\,x^3-x^2+3\right)J_{0}(x)\,J_1(x)^2+4\,\left(4\,x^3-x^2+5\,x+1
\right)J_1(x)^3\right)
\end{eqnarray*}

\noindent
Similarly

$ Y_2(x)=I_{0}(x)^2-I_1(x)^2 $

$
Y_3(x)=x^{-1}\left(2\,\sqrt{x}\,I_{0}(x)^2\,I_1(x)-I_{0}(x)\,I_1(x)^2-2\,\sqrt{x}\,I_1(x)^3\right)
$

\begin{eqnarray*}
Y_4(x) &amp;=&amp; x^{-3}\big(-4\,x^2\,I_{0}(x)^4
+8\,x\,\sqrt{x}\,I_{0}(x)^3\,I_1(x)
+4\,x\,(2\,x-1)\,I_{0}(x)^2\,I_1(x)^2 \\
&amp; \qquad &amp;
-8\,x\,\sqrt{x}\,I_{0}(x)\,I_1(x)^3
-x\,(4\,x-1)\,I_1(x)^4\big)
\end{eqnarray*}

A theoretical argument (see \cite{G})  shows that the generating
functions $y_k(x)$ and $Y_k(x)$  are {\sl $D$-finite\/}.
That is to
say, they satisfy linear differential equations with polynomial
coefficients. In fact, it is well known and classical that one can
translate such linear differential equations into recurrences with
polynomial coefficients.  More precisely, a {\sl $P$-recurrence\/}
for a sequence
$a_n$ is one of the form
  $$p_0(n)\,a_n+p_1(x)\,a_{n-1}+\ldots+p_k(n)\,a_{n-k}=q(n),$$
where all $p_i(n)$, $1\leq i\leq k$, and $q(n)$ are polynomials in $n$.
We say that a sequence is $P$-recursive if it satisfies a
$P$-recurrence. The
class of $P$-recursive sequences is closed under point-wise products.
Since $1/n!$ is easily seen to be $P$-recursive, it follows that, if
$a_n$ is
$P$-recursive, then so are $a_n/n!$ and $a_n/n!^2$.  The algorithmic
translation from $D$-finite to $P$-recursive (and back) has been
implemented in the
package GFUN in Maple (see \cite{GFUN}), which also contains many other nice tools
for handling recurrences and generating functions.

Computer experiments made by Krob, Favreau and the first author led to
conjectures (see
\cite{BFK}) for an explicit form for $P$-recurrences for $\tau_h(n)$ and
$\Tau_h(n)$. These conjectures can be easily (and automatically)
reformulated as
linear differential equations for $y_k(x)$ and $Y_k(x)$. We first
observe that it is not hard to show the existence of a
linear differential equation of order bounded by

  $$\ell(k):=\floor{k\over 2}+1$$
with polynomial coefficients,
admitting $y_k(x)$ as a solution. In fact, this follows readily from
the following proposition.

\noindent {\bf Proposition 1.} \ {\sl Let  ${\cal V}_k$  denote the
vector space over the field $\C(x)$ of rational functions in $x$
spanned by
$y_k(x)$ and all its derivatives. Then\/}
  $$\dim\,{\cal V}_k\leq \ell(k).$$

\begin{proof}
Setting $n:=\ell(k)-1$, it is clear from our
previous discussion that $y_k$ lies in the span ${\cal W}_k$ of the set
of
$\ell(k)$ elements given by
  $$\{J_0(x)^m\,J_1(x)^{n-m}\ |\ 0\leq m\leq n\ \}$$
if $k$ is even, and by
  $$\{e^x\,J_0(x)^m\,J_1(x)^{n-m}\ |\ 0\leq m\leq n\ \}$$
if $k$ is odd.
${\cal W}_k$ is clearly closed under differentiation,
since we easily see that
 \begin{eqnarray} 
     {d\over dx} J_0(x)&amp;= &amp; 2\,J_1(x), \nonumber\\
     {d\over dx} J_1(x)&amp;= &amp; 2\,J_0(x)-{1\over x}\,J_1(x), \label{eq1}
\end{eqnarray}
from which we deduce that
   $${d\over dx} J_0(x)^a\,J_1(x)^b= {2\,a \over
x}\,J_{0}(x)^{a-1}\,J_1(x)^{b+1}\,x
                      +2\,b\,J_{0}(x)^{a+1}\,J_1(x)^{b-1}-{b\over
x}\,J_{0}(x)^a\,J_1(x)^b ,\eqno{(2)}$$
as well as a similar expression for the derivative of
$e^x\,J_0(x)^a\,J_1(x)^b$.
Thus ${\cal V}_k$ is contained in ${\cal
W}_k$, and hence its
dimension is bounded by $\ell(k)$.
\end{proof}

Setting for the moment $n:=\ell(k)$ and $y:=y_k(x)$, it clearly follows
from the above proposition that
   $$y,\ y',\ y'',\ \ldots\ , y^{(n)}$$
are linearly dependent, hence $y_k(x)$ satisfies a homogeneous linear
differential equation of order (at most) $\ell(k)$
with polynomial coefficients (in $x$). 
However, it appears that a stronger result holds.

\noindent {\bf Conjecture} (Bergeron-Favreau-Krob, \cite{BFK}).
For each $k$,
there are polynomials $p_m(x)$ of degree at
most $\ell-1$ such that $y_k(x)$  is a solution of
  $$\sum_{m=0}^{\ell} p_m(x) y^{(m)}\ =\ 0,$$
where $\ell=\ell(k)$. Moreover, for $m\geq 1$, $p_m(x)=q_m(x)\,x^{m-1}$,
and $p_\ell(x)=x^{\ell-1}$.

\noindent The first few cases for $y_k(x)$ are\footnote{$^*$}{
Here $\rightarrow$ means ``is a solution of''.}

\noindent $y_2(x)\ \rightarrow\quad  x\,y''+2\,y'-2\,(2\,x+1)\,y=0 $

\noindent $y_3(x)\ \rightarrow\quad   x\,y''-(2\,x-3)\,y'-3\,(x+1)\,y=0
$

\noindent $y_4(x)\ \rightarrow\quad
x^2\,y'''+10\,x\,y''-4\,\left(4\,x^2+2\,x-5\right)y'-4\,(8\,x+5)\,y=0 $

\noindent $y_5(x)\ \rightarrow\quad   x^2\,y'''-(3\,x-13)\,x\,y''
-\left(13\,x^2+26\,x-35\right)y'+5\,\left(3\,x^2-7\,x-7\right)y=0 $

\noindent
Equating coefficients of $x^n/n!$ on both hand sides of these
differential equations, one finds that they are equivalent to the
recurrences

\noindent $ (n+1)\,\tau_2(n)-2\,\tau_2(n-1)-4\,(n-1)\,\tau_2(n-2)=0 $

\noindent $
(n+2)\,\tau_3(n)-(2\,n+1)\,\tau_3(n-1)-3\,(n-1)\,\tau_3(n-2)=0 $

\noindent $  (n+3)\,(n+4)\,\tau_4(n){-}16\,(n-1)\,\tau_4(n-2)\,n-(8
\,n+12)\,\tau_4(n-1)=0  $

\noindent $ (n+4)\,(n+6)\,\tau_5(n)-\left(3\,n^2+17\,n+15\right)\tau
_5(n-1)-(n-1) (13\,n+9)\,\tau_5(n-2)+15\,(n-1)\,(n-2)\,\tau_
5(n-3)=0  $




Up to now, only these recurrences (that is, for $k\leq 5$), had been
implicitly known (see \cite{GB}). However,
using the simplified expressions for  $y_k(x)$ given here, and a
reformulation in term of linear differential equations (with the help of
GFUN
(\cite{GFUN}) we have been able to  check (in the form of a computer algebra
proof) that the conjecture above is true for $k\leq 11$, from which it follows
that
the corresponding recurrences hold. This computer verification simply
uses the derivation rules (\ref{eq1}) for $J_0(x)$ and $J_1(x)$
to simplify the expressions obtained by substitution of Gessel's formulae
in the following differential equations.

\begin{eqnarray*}
y_6(x) &amp;\rightarrow\quad &amp; x^3\,y^{(4)}+28\,x^2\,y'''
-10\,\left(4\,x^2+2\,x-23\right)x\,y''\\
&amp;\quad &amp; -4\,\left(108\,x^2+61\,x-135\right)y'
+36\,(2\,x+5)\left(2\,x^2-3\,x-3\right)y=0;
\end{eqnarray*}

\begin{eqnarray*}
y_7(x) &amp; \rightarrow\quad &amp;
x^3\,y^{(4)}-2\,(2\,x-17)\,x^2\,y'''-
\left(34\,x^2+102\,x-343\right)x\,y''  \\
&amp; \quad &amp; +\left(76\,x^3-450\,x^2-686\,x+1001\right)y'
+7\,\left(15\,x^3+74\,x^2-143\,x-143\right)y=0.
\end{eqnarray*}

\begin{eqnarray*}
y_8(x) &amp;\rightarrow\quad &amp; x^4\,y^{(5)}+60\,x^3\,y^{(4)}
-2\,\left(40\,x^2+20\,x-619\right)x^2\,y'''
-4\,\left(608\,x^2+331\,x-2567\right)x\,y''  \\
&amp; \quad &amp; +8\,\left(128\,x^4+128\,x^3-2480\,x^2-1527\,x+3536\right)y' \\
&amp; \quad &amp; +128\,\left(64\,x^3+72\,x^2-286\,x-221\right)y=0;
\end{eqnarray*}

\begin{eqnarray*}
y_9(x) &amp; \rightarrow\quad &amp;
x^4\,y^{(5)}-5\,(x-14)\,x^3\,y^{(4)}
-\left(70\,x^2+280\,x-1693\right) x^2\,y'''
\\
&amp;\quad&amp; +\left(230\,x^3-2492\,x^2-5079\,x+16535\right) x\,y''
\\
&amp;\quad &amp; +\left(789\,x^4+5544\,x^3-24073\,x^2-33070\,x+53865\right)y'  \\
&amp;\quad &amp; -27\,\left(35\,x^4-274\,x^3-1017\,x^2+1995\,x+1995\right)y=0;
\end{eqnarray*}

\begin{eqnarray*}
y_{10}(x) &amp;\rightarrow\quad &amp; x^5\,y^{(6)}+110\,x^4\,y^{(5)}
-2\,\left(70\,x^2+35\,x-2269\right)x^3\,y^{(4)}   \\
&amp;\quad &amp; -4\,\left(2268\,x^2+1211\,x-21752\right)x^2\,y''' \\
&amp;\quad &amp; +4\,\left(1036\,x^4+1036\,x^3-48033\,x^2-27900\,x+191477\right)x\,y''
\\
&amp; \quad &amp; +8\,\left(14300\,x^4+15542\,x^3-185404\,x^2-121352\,x+303875\right)y'
\\
&amp; \quad &amp; -200\,\left(72\,x^5+108\,x^4-3262\,x^3-3987\,x^2+14960\,x+12155\right)y=0;
\end{eqnarray*}

\begin{eqnarray*}
y_{11}(x) &amp; \rightarrow\quad &amp;
x^5\,y^{(6)}-(6\,x-125)\,x^4\,y^{(5)}-\left(125\,x^2+625\,x-5873\right)
x^3\,y^{(4)}  \\
&amp;\quad &amp; +2\,\left(270\,x^3-4611\,x^2-11746\,x+64252\right)x^2\,y''' \\
&amp;\quad &amp; +\left(3319\,x^4+30166\,x^3
-223422\,x^2-385512\,x+1293125\right)x\,y'' \\
&amp;\quad &amp; -\left(7734\,x^5-104329\,x^4-493828\,x^3+1987124\,x^2+2586250\,x-4697275\right)y' \\
&amp; \quad &amp; 
-11\,\left(945\,x^5+11343\,x^4-62023\,x^3
-204012\,x^2+427025\,x+427025\right)y=0 .
\end{eqnarray*}

\noindent However, these verifications rapidly become (computer) time
consuming. For example, with $k=11$, we have to substitute in this last
differential equation the following expression

\begin{eqnarray*}
y_{11}(x) &amp;= &amp; {138240\,e^{x}\over
x^{25}} \Big(-14\,{\left(32\,x^6+177\,x^4+198\,x^2-72\right)x^5\,J_{0}(x)^5} \\
&amp;\quad &amp; 
+8\,\left(16\,x^8+256\,x^6+825\,x^4+585\,x^2-495\right)x^4\,J_1(x)\,J_{0}(x)^4 \\
&amp; \quad &amp;
+4\,\left(192\,x^8+833\,x^6+495\,x^4+135\,x^2+1440\right)x^3\,J_1(x)^2\,J_{0}(x)^3 \\
&amp; \quad &amp; 
-\left(256\,x^{10}+3648\,x^8+10799\,x^6+9690\,x^4+1980\,x^2+3600\right)x^2\,J_1(x)^3\,J_{0}(x)^2 \\
&amp; \quad &amp; 
-5\,\left(64\,x^{10}+190\,x^8-77\,x^6+114\,x^4+504\,x^2-144\right)x\,J_1(x)^4\,J_{0}(x) \\
&amp; \quad &amp; 
+\left(128\,x^{12}+1632\,x^{10}+4557\,x^8+5482\,x^6+4158\,x^4+2052\,x^2+72\right)J_1(x)^5\Big)
\end{eqnarray*} 

\noindent
and simplify. Clearly we could go on to larger cases, but the point
seems to be made that the conjectures are reasonable.


\noindent Similar considerations for the enumeration of pairs of
tableaux, with the following differential equations, settle the
corresponding conjectures
for the cases
$k\leq 7$:

\noindent $ Y_2(x)\ \rightarrow\quad
x^2\,y'''+4\,x\,y''-2\,(2\,x-1)\,y'-2\,y=0 $


\noindent $ Y_3(x)\ \rightarrow\quad
x^3\,y^{(4)}
+10\,x^2\,y'''
-(10\,x-23)\,x\,y''
-(32\,x-9)\,y'
+9\,(x-1)\,y=0$


\begin{eqnarray*}
Y_4(x) &amp; \rightarrow &amp; 
x^4\,y^{(5)}
+20\,x^3\,y^{(4)}
-2\,(10\,x-59)\,x^2\,y'''
-2\,(91\,x-110)\ x\,y'' \\
&amp; \quad &amp; 
+4\,\left(16\,x^2-87\,x+20\right)y'
+16\,(8\,x-5)\,y=0.
\end{eqnarray*}


\begin{eqnarray*}
Y_5(x) &amp;\rightarrow\quad &amp;
x^5\,y^{(6)}
+35\,x^4\,y^{(5)}
-7\,(5\,x-59)\,x^3\,y^{(4)}
-2\,(336\,x-979)\, x^2\,y''' \\
&amp; \quad &amp; +\left(259\,x^2-3650\,x+3383\right)x\,y'' \\
&amp; \quad &amp; + \left(1917\,x^2-5708\,x+1225\right)y'
-25\,\left(9\,x^2-93\,x+49\right)y=0.
\end{eqnarray*}

\begin{eqnarray*}
Y_6(x) &amp;\rightarrow\quad &amp;
x^6\,y^{(7)}
+56\,x^5\,y^{(6)}
-28\,(2\,x-41)\,x^4\,y^{(5)}
-4\,(483\,x -2684)\,x^3\,y^{(4)} \\
&amp;\quad &amp; 
+4\,\left(196\,x^2-5480\,x+11543\right)x^2\,y'''
+8\,\left(1686\,x^2-11941\,x +9830\right)x\,y'' \\
&amp; \quad &amp;
-4\,\left(576\,x^3-14931\,x^2+34438\,x-7290\right)y'
-72\,\left(144\,x^2-821\,x+405\right)y=0 .
\end{eqnarray*}


\begin{eqnarray*}
Y_7(x) &amp; \rightarrow\quad &amp;
x^7\,y^{(8)}
+84\,x^6\,y^{(7)}
-42\,(2\,x-65)\,x^5\,y^{(6)}
-2\,(2352\,x -21881)\,x^4\,y^{(5)} \\
&amp;\quad &amp; +3\,\left(658\,x^2-31606\,x+121455\right)x^3\,y^{(4)} \\
&amp; \quad &amp; +2\,\left(31986\,x^2-424260\,x+754183\right)x^2\,y''' \\
&amp; \quad &amp; -\left(12916\,x^3-648834\,x^2+3329230\,x-2610671\right)x\,y''  \\
&amp; \quad &amp; -\left(175704\,x^3-2292734\,x^2+4684008\,x-1002001\right)y' \\
&amp; \quad &amp; +49\,\left(225\,x^3-9630\,x^2+42313\,x-20449\right)y=0
.
\end{eqnarray*}


\section{Acknowledgments}
The Maple package {\sl gfun\/} (available as a shared library) was used
extensively
in the elaboration of the conjectures and results in this note.

\begin{thebibliography}{9}

\bibitem{BFK} F.~Bergeron, L.~Favreau and D.~Krob, {\sl Conjectures on the
Enumeration  of Tableaux of Bounded Height\/}, {  Discrete
Math.}, {\bf 139}, (1995), 463--468.


\bibitem{G} I.~Gessel, {\sl Symmetric Functions and P-Recursiveness\/},
Jour. of Comb. Th., Series A, {\bf 53}, 1990, 257--285.

\bibitem{GB} D.~Gouyou Beauchamps, {\sl Codages par des mots et des chemins:
probl\`emes combinatoires et
algorithmiques\/}, Ph. D. thesis, University of Bordeaux I, 1985.

\bibitem{GFUN} B.~Salvy and P.~Zimmermann, {\sl GFUN: A maple Package for the
Manipulation of Generating Functions in one
Variable\/}, ACM Trans. in Math. Software, {\bf 20}, 1994, pages
163--177.

\bibitem{eis}
N. J. A.~Sloane, The On-Line Encyclopedia of Integer Sequences,
published electronically at
\url{http://oeis.org}.
See also N. J. A.~Sloane and S.~Plouffe, {\sl The Encyclopedia of Integer
Sequences\/}, Academic Press, 1995.

\end{thebibliography}

\vskip 1cm
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A000108},
\seqnum{A001006},
\seqnum{A001405},
\seqnum{A005802},
\seqnum{A005817},
\seqnum{A007579},
\seqnum{A049401},
\seqnum{A052397},
\seqnum{A052398}, and
\seqnum{A052399}.)

\bigskip
\hrule
\bigskip

\noindent Received November 10, 1999;
published in Journal of Integer Sequences March 15, 2000.

\bigskip
\hrule
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\noindent Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in

\bigskip
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