On odd ranks of odd Durfee symbols

Abstract

An odd Durfee symbol of [Formula: see text] is an array of positive odd integers and a subscript [Formula: see text], [Formula: see text] such that [Formula: see text], [Formula: see text], and [Formula: see text]. Andrews defined the odd rank of an odd Durfee symbol as [Formula: see text]. Let [Formula: see text] be the number of odd Durfee symbols of [Formula: see text] with odd rank congruent to [Formula: see text] modulo [Formula: see text]. We decompose the generating function of [Formula: see text] into modular and mock modular parts. Specifically, we derive some special cases of the generating functions of [Formula: see text] for [Formula: see text]. Some generating functions are related to classical mock theta functions.