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Is there some reference for sums like: $$\sum_{\nu \subset \mathrm{[1,n] x[1, m]}}s_{\nu}(x)s_{\nu}(y)t^{|\nu|}$$ $$\sum_{\nu \subset \mathrm{[1,n] x[1, m]}}s_{\nu}(x)s_{\nu}(y)\cdot|\nu|$$ (summation over diagrams lying inside the box of length $m$ and width $n$) or specializations with all $x_i = y_j = 1$ ? Any of these will be sufficient for me.

edit: $s_{\nu}(x)$ depends on $x_1, ..., x_n$, while $s_{\nu}(y)$ on any number of variables.

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  • $\begingroup$ what are the cardinalities of the alphabets? n and m, or something else? $\endgroup$
    – user35313
    Sep 19, 2019 at 9:01
  • $\begingroup$ @user61318 cardinalities are: $n$ for $x$'s and any for $y$'s. Sorry for not specifying it. $\endgroup$ Sep 19, 2019 at 9:37
  • $\begingroup$ The Cauchy-identities goes hand in hand with RSK, math.upenn.edu/~peal/polynomials/schur.htm#schurCauchyFormula and math.upenn.edu/~peal/polynomials/rsk.htm#rsk See the references given there for proofs. The second identity should follow from the first by taking derivative, and setting t=1.. $\endgroup$ Sep 19, 2019 at 10:28
  • $\begingroup$ The first one should be a sum over integer matrices, such that the corresponding biwords (see RSK) has some restriction on longest increasing subsequence, and |t| is simply sum of entries in the matrix... $\endgroup$ Sep 19, 2019 at 10:43

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