Reciprocals of Subsum Polynomials

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