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%           FPSAC/SFCA '94, May 23-27, 1994   at DIMACS
%                     Problem Session
%   LaTeX file of problem  #1  submitted by David M. Jackson (May 24, 1994)
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{FPSAC/SFCA '94}\\
{PROBLEM SESSION}\\
{Submission received May 24, 1994}\\[.2in]
{\bf A GENERALIZED CAUCHY IDENTITY}\\[.1in]
David M.\ Jackson\\[.2in]
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Find a bijective proof of the identity
  $$ \sum_{\theta} \sum_{1\leq i_1<\cdots <i_{k-1}<n} (\theta_{i_1}+k-1)
     (\theta_{i_2}+k-2)\cdots (\theta_{i_{k-1}}+1)\cdot $$
  $$ \quad (\theta_{i_{k-1}+1}+\cdots +\theta_n)s_{\theta}(y_1,\dots,
     y_n)s_\theta(w_1,\dots,w_m) $$
  $$ = \prod_{i=1}^{n}\prod_{j=1}^{m}\frac{1}{1-y_iw_j}\sum_{j_1<
    \cdots <j_k\atop i_1\neq\cdots\neq i_k}\prod_{l=1}^{k}
    \frac{y_{j_l}w_{i_l}}{1-y_{j_l}w_{i_l}}. $$
Here, $\theta$ is a partition $(\theta_1,\theta_2,\ldots),$ and
$s_\theta$ is a Schur function.
\noindent See {\em Trans.\ Amer.\ Math.\ Soc.}, {\bf310} (1988), 805--820.
The identity appears on p.\ 809.

It suggests that the Robinson-Schensted algorithm can be extended by
marking boxes in the first row in a special way.
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