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Hook lengths and 3-cores

By Guo-Niu Han and Ken Ono

[download the paper: .ps - .pdf, 7 pages, 2008/05/12]

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 A000731(n)=0 if and only if A033687(n)=0 (Conjecture 4.6 in [4]). The numbers A000731(n) are given in terms of hook lengths of partitions, while A033687(n) equals the number of 3-core partitions of n. Here we prove this conjecture.

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