Congruences for Han’s generating function

Collins, Dan; Wolfe, Sally

doi:10.2140/involve.2009.2.225

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Congruences for Han's generating function

Dan Collins and Sally Wolfe

Vol. 2 (2009), No. 2, 225–236

DOI: 10.2140/involve.2009.2.225

Abstract

For an integer $t \geq 1$ and a partition $λ$ , we let $ℋ_{t} (λ)$ be the multiset of hook lengths of $λ$ which are divisible by $t$ . Then, define $a_{t}^{e v e n} (n)$ and $a_{t}^{o d d} (n)$ to be the number of partitions of $n$ such that $| ℋ_{t} (λ) |$ is even or odd, respectively. In a recent paper, Han generalized the Nekrasov–Okounkov formula to obtain a generating function for $a_{t} (n) = a_{t}^{e v e n} (n) - a_{t}^{o d d} (n)$ . We use this generating function to prove congruences for the coefficients $a_{t} (n)$ .

Keywords

partition, partition function, Han's generating function, Nekrasov–Okounkov, hook length, Ramanujan congruences, congruences, modular forms

Mathematical Subject Classification 2000

Primary: 05A17, 11P83

Milestones

Received: 29 September 2008

Accepted: 17 January 2009

Published: 7 May 2009

Communicated by Kenneth S. Berenhaut

Authors

Dan Collins
Department of Mathematics Cornell University 310 Malott Hall Ithaca, NY 14853 United States

Sally Wolfe
Department of Mathematics University of Wisconsin Madison, WI 53706 United States

Vol. 2, No. 2, 2009