Partitions with parts separated by parity: conjugation, congruences and the mock theta functions

TL;DR

This paper explores partitions with parts separated by parity, establishing new combinatorial identities, congruences, and connections to third order mock theta functions through conjugation-invariant subsets.

Contribution

It introduces novel involutions, proves generalized identities, and links partition subsets to Ramanujan's mock theta functions using combinatorial methods.

Findings

A Franklin-type involution for restricted partitions.

New congruences of Andrews--Beck type.

Connections between stable partition subsets and mock theta functions.

Abstract

Noting a curious link between Andrews' even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is twofold. First off, we derive results for certain restricted partitions with even parts below odd parts. These include a Franklin-type involution proving a parametrized identity that generalizes Andrews' bivariate generating function, and two families of Andrews--Beck type congruences. Secondly, we introduce several new subsets of partitions that are stable (i.e., invariant under conjugation) and explore their connections with three third order mock theta functions $\omega(q)$, $\nu(q)$, and $\psi^{(3)}(q)$, introduced by Ramanujan and Watson.