Untrodden pathways in the study of partitions with parts separated by parity

Abstract

In this paper, we study a partition function with parts separated by parity from a family of partition functions introduced by Andrews in 2019. We first show that its generating function admit representation in terms of mock modular forms of weight $3/2$. We then initiate a study of smallest-part partition function analogues of partitions with parts separated by parity, which, to the best of our knowledge, has not previously been investigated in the literature. We show that the generating functions of these smallest-part partition functions are likewise expressible in terms of mock modular forms, mock Maass theta functions and Zagier's $q$-series $H(q)$, which he describes as a ``half-derivative of the Dedekind eta-function". Our study also leads to new combinatorial interpretations of the coefficients of the aforementioned mock modular form, function $σ^*(q)$, and mock theta functions $ψ(q)$ and $A(q)$ in terms of odd Ferrers graphs. Moreover, we provide a quantum $q$-series identity involving Ramanujan's famous $σ(q)$ function. We also derive the leading asymptotic term of one of these partition functions, whose generating function we first express in terms of $σ^*(q)$.