# Restricted Cauchy identity

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.

• what are the cardinalities of the alphabets? n and m, or something else?
– user35313
Sep 19, 2019 at 9:01
• @user61318 cardinalities are: $n$ for $x$'s and any for $y$'s. Sorry for not specifying it. Sep 19, 2019 at 9:37
• 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.. Sep 19, 2019 at 10:28
• 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... Sep 19, 2019 at 10:43