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A Conjecture of Han on 3-Cores and Modular Forms

Author

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  • Amanda Clemm

    (Department of Mathematics, Emory University, Emory, Atlanta, GA 30322, USA)

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.

Suggested Citation

  • Amanda Clemm, 2014. "A Conjecture of Han on 3-Cores and Modular Forms," Mathematics, MDPI, vol. 2(4), pages 1-8, December.
    • Handle: RePEc:gam:jmathe:v:2:y:2014:i:4:p:232-239:d:43746
    as

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