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Mathematics 2014, 2(4), 232-239; doi:10.3390/math2040232

A Conjecture of Han on 3-Cores and Modular Forms

Department of Mathematics, Emory University, Emory, Atlanta, GA 30322, USA
Received: 24 October 2014 / Revised: 16 December 2014 / Accepted: 17 December 2014 / Published: 19 December 2014
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Abstract

In his study of Nekrasov–Okounkov type formulas on “partition theoretic” expressions for families of infinite products, Han discovered seemingly unrelated q-series that are supported on precisely the same terms as these infinite products. In collaboration with Ono, Han proved one instance of this occurrence that exhibited a relation between the numbers a(n) that are given in terms of hook lengths of partitions, with the numbers b(n) that equal the number of 3-core partitions of n. Recently Han revisited the q-series with coefficients a(n) and b(n), and numerically found a third q-series whose coefficients appear to be supported on the same terms. Here we prove Han’s conjecture about this third series by proving a general theorem about this phenomenon. View Full-Text
Keywords: hook length; partition; modular form hook length; partition; modular form
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Clemm, A. A Conjecture of Han on 3-Cores and Modular Forms. Mathematics 2014, 2, 232-239.

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