Congruences for Han's generating function

Abstract

For an integer $t\ge 1$ and a partition $\lambda$, we let ${\mathcal{\mathscr{H}}}_t\left(\lambda \right)$ be the multiset of hook lengths of $\lambda$ which are divisible by $t$. Then, define $a_t^{even}\left(n\right)$ and $a_t^{odd}\left(n\right)$ to be the number of partitions of $n$ such that $\left|{\mathcal{\mathscr{H}}}_t\left(\lambda \right)\right|$ is even or odd, respectively. In a recent paper, Han generalized the Nekrasov–Okounkov formula to obtain a generating function for $a_t\left(n\right)=a_t^{even}\left(n\right)-a_t^{odd}\left(n\right)$. We use this generating function to prove congruences for the coefficients $a_t\left(n\right)$.