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A Conjecture of Han on 3-Cores and Modular Forms

Department of Mathematics, Emory University, Emory, Atlanta, GA 30322, USA
Mathematics 2014, 2(4), 232-239; https://doi.org/10.3390/math2040232
Received: 24 October 2014 / Revised: 16 December 2014 / Accepted: 17 December 2014 / Published: 19 December 2014
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
MDPI and ACS Style

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

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