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Combinatorial reciprocity for symmetric functions

Suppose we have two sets of variables, $\{x_i\}_{i\geq 1},\{y_i\}_{i\geq 1}$ and we evaluate the symmetric function $f$ with their pairwise products as inputs: $$f(x_1y_1,x_1y_2,...,x_iy_j,...).$$ ...

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