Vol. 2, No. 2, 2009

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Congruences for Han's generating function

Dan Collins and Sally Wolfe

Vol. 2 (2009), No. 2, 225–236
Abstract

For an integer $t\ge 1$ and a partition $\lambda$, we let ${\mathsc{ℋ}}_{t}\left(\lambda \right)$ 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 $|{\mathsc{ℋ}}_{t}\left(\lambda \right)|$ 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 ${a}_{t}\left(n\right)$.

Keywords
partition, partition function, Han's generating function, Nekrasov–Okounkov, hook length, Ramanujan congruences, congruences, modular forms
Mathematical Subject Classification 2000
Primary: 05A17, 11P83