Asymptotics of parity biases for partitions into distinct parts via Nahm sums

Abstract

For a random partition, one of the most basic questions is: what can one expect about the parts which arise? For example, what is the distribution of the parts of random partitions modulo $N$? Since most partitions contain a $1$, and indeed many $1$s arise as parts of a random partition, it is natural to expect a skew towards $1(\mathrm{mod}N)$. This is indeed the case. For instance, Kim, Kim, and Lovejoy recently established ``parity biases'' showing how often one expects partitions to have more odd than even parts. Here, we generalize their work to give asymptotics for biases $\mathrm{mod}N$ for partitions into distinct parts. The proofs rely on the Circle Method and give independently useful techniques for analyzing the asymptotics of Nahm-type $q$-hypergeometric series.

31 pages, 4 figures