diagnosis journal

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.