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Hook Lengths and 3-Cores
Guo-Niu Han1 and Ken Ono2
1Institut de RechercheMathématique Avancée , UMR 7501, Université Louis Pasteur et CNRS, 7 rue Ren´e-Descartes, F-67084 Strasbourg, France
guoniu@math.u-strasbg.fr
2Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USA
ono@math.wisc.edu
Annals of Combinatorics 15 (2) pp.305-312 April, 2011
AMS Subject Classification: 05A17, 05A30, 11D09, 11F11
Abstract:
Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he conjectured that a(n) = 0 if and only if b(n) = 0, where integers a(n) and b(n) are defined by

The numbers a(n) are given in terms of hook lengths of partitions, while b(n) equals the number
of 3-core partitions of n. Here we prove this conjecture.
Keywords: hook length, t-core, partition, modular form

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