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\begin{document}
\title{A bijective proof of Lassalle's partition identity}
\author{ Jiang Zeng\\
\small Institut Girard D\'esargues, Universit\'e Claude Bernard Lyon 1\\
\small  43, boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex\\
\small e-mail: zeng@desargues.univ-lyon1.fr}
%\date{June 8, 1999}
\maketitle
We give a bijective proof of the following form of
a conjecture in~\cite{lass1,lass2}. The special $s=1$ case was proved in the same vein by 
Simion~\cite{lass1}. For other proofs of this conjecture, see
\cite{lass3}.
Our notations are the same as in 
\cite{lass1,lass2}.  For an integer 
partition $\mu$ let the length $l(\mu)$ be the number of
the parts of $\mu$, $m_i(\mu)$ the number of parts equal to
$i$, $z_\mu=\prod_{i\geq 1}i^{m_i(\mu)}m_i(\mu)!$ and
$\langle{\mu\atop r}\rangle$ the number of ways to choose $r$
different cells from the diagram of the partition $\mu$ taking at
least one cell from each row. For any positive integer $n$ let
$(a)_n=a(a+1)\cdots (a+n-1)$ and $[n]=\{1,2,\ldots, n\}$.

\begin{theorem} We have
\begin{eqnarray}\label{conj1}
\sum_{|\mu|=n}\frac{n!}{z_\mu}\langle{\mu\atop r}\rangle
X^{l(\mu)-1}
&&\hskip -1cm\sum_{i=1}^{l(\mu)}\mu_i{\mu_i+s-1\choose s-1}\nonumber\\
&=&{n\choose r}(s+r)_{n-r}\left[(X+s)_r-(X)_r\right].
\end{eqnarray}
\end{theorem}

Recall that the functional graph of an injection $f :[n]\to [n+r]$
consists of disjoint cycles and paths. Each path is of form 
$a_0\to a_1\to \cdots \to a_l$, where $f(a_i)=a_{i+1}$ for $0\leq j<l$, 
with $f^{-1}(a_0)$ empty, and $a_l\in [n+r]\setminus [n]$.
Each cycle is of the form $a_0\to a_1\to \cdots \to a_l$, 
where $f(a_i)=a_{i+1}$ for $0\leq j\leq l$, 
with $a_{l+1}=a_0$.
 Let $\cyc f$ be
 the number of cycles in $f$. Then the corresponding 
generating function  reads as follows:
\begin{equation}
\sum_{f:[n]\to [n+s]}X^{\cyc f}=(X+s)_n
\end{equation} 
where $f$ is an injection. The above identity seems to be 
 first published by Foata and Strehl \cite{FoSt} and  can be readily verified 
by induction on $n$ as in the well-known  
case of permutations, which corresponds to $s=0$.
 
We shall prove that the two sides of (\ref{conj1}) are the generating
function of pairs $(f,\,R)$ with respect to the number of cycles of $f$, 
where $f$ is an 
injection from $[n]$
to $[n+s]$ and $R$ is a $r$-subset of $[n]$ such that each
cycle of $f$ has at least one element in $R$ and the restriction of $f$ on $R$ is not a permutation of $R$.

Indeed, to construct such a pair $(f,\, R)$ we can first 
choose $r$ elements of $[n]$ to form a $r$-subset $R$, there are 
${n\choose r}$  ways to do so; and construct an injection 
$g:R\to R\cup \{n+1,\ldots, n+s\}$ such that its restriction on $R$ is 
 not a permutation
of $R$, the generating function of such injections is then
$(X+s)_r-(X)_r$; finally insert  succesively the $n-r$
elements of $[n]\setminus R$ into the above injection $g$ to 
obtain the injection $f: [n]\to [n+s]$, there are
 $(s+r)_{n-r}$  ways to insert the $n-r$
elements into any above injection $g$. This gives the right-hand side of
(\ref{conj1}).

On the other hand, we can construct such a pair $(f,\,R)$ as follows:
construct first a permutation $\sigma$ 
on $[n]$, for any partition $\mu$ of $n$ 
there are $n!/z_\mu$
such permutations of type $\mu$;
choose $r$ elements of $[n]$ to form the subset $R$ 
such that each cycle of $\sigma$ has at least one 
element in $R$, there are $\langle{\mu\atop r}\rangle$ ways to do so;
choose any  cycle $(a,\,\sigma(a),\,\cdots \,,\sigma^{\mu_i-1}(a))$ 
of $\sigma$ with length $\mu_i$ to form
$s$ paths:
\begin{eqnarray*}
a\to \sigma(a)\to \cdots \sigma^{l_1-1}(a)&\to& n+1,\\
\sigma^{l_1}(a)\to \sigma^{l_1+1}(a)\to \cdots \sigma^{l_2-1}(a)&\to& n+2,\\
\cdots\cdots&\to &\cdots\\
\sigma^{l_{s-1}}(a)\to \sigma^{l_{s-1}+1}(a)\to \cdots \sigma^{l_s-1}(a)&\to& n+s,
\end{eqnarray*}
where $l_1+l_2+\cdots l_s=\mu_i$ and by convention  the
corresponding path is empty if $l_i=0$.
Obviously there are
$\mu_i{\mu_i+s-1\choose s-1}$ ways to form  such $s$  paths starting from any
cycle of length $\mu_i$.
Finally we complete the construction  of $f:[n]\to [n+s]$
by using the $s$ paths above and the remaining $l(\mu)-1$ cycles of $\sigma$.
This gives the left-hand side of (\ref{conj1}).

The theorem follows then from the two above countings.


\begin{thebibliography}{99}

\bibitem{FoSt} \textsc{Foata} (D.) and \textsc{Strehl} (V.)
\emph{Combinatorics of Laguerre polynomials}, in~:
Enumeration and Design, Waterloo Jubilee Conference (Academic Press,
New York, 1984) 123--140.

\bibitem{lass1} \textsc{Lassalle} (M.) \emph{Quelques conjectures combinatoires
relatives \`a la formule classique de Chu-Vandermonde}, Adv. in
Appl. Math.
21 (1998), 457-472.

\bibitem{lass2}\textsc{Lassalle} (M.) \emph{Une conjecture en
th\'eorie des partitions},
http://xxx.lanl.gov/math.CO/9901040, 1999.

\bibitem{lass3}\textsc{Lassalle} (M.) \emph{Une identit\'e  en
th\'eorie des partitions},
Journal of Combinatorial Theory, Series A, to appear.
\end{thebibliography}

\end{document}
