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 1 and a partition λ, we let t(λ) be the multiset of hook lengths of λ which are divisible by t. Then, define ateven(n) and atodd(n) to be the number of partitions of n such that |t(λ)| is even or odd, respectively. In a recent paper, Han generalized the Nekrasov–Okounkov formula to obtain a generating function for at(n) = ateven(n) atodd(n). We use this generating function to prove congruences for the coefficients at(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