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.