A Conjecture of Han on 3-Cores and Modular Forms - Amanda Clemm (Mathematics, 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.