Reciprocals of Subsum Polynomials

Abstract

We introduce the subsum polynomial of a partition $\hat{I}»=(\hat{I}{»}_1,\hat{I}{»}_2,\dots ,\hat{I}{»}_k)$ defined by $\mathrm{sp}(\hat{I}»,x)={\prod }_{i=1}^k(1+x^{\hat{I}{»}_i})$. We study the sum of reciprocals of $\mathrm{sp}(\hat{I}»,x)$ over all partitions of $n$. We prove arithmetic properties of related polynomials and offer connections to other combinatorial objects.

23 pages