Combinatorial Proofs of Two Overpartition Theorems Connected by a Universal Mock Theta Function

TL;DR

This paper provides combinatorial proofs for two overpartition theorems linked by a universal mock theta function, expanding understanding of overpartition identities and their generating functions.

Contribution

It introduces combinatorial methods to derive generating functions for overpartition theorems connected to universal mock theta functions, complementing previous q-differential equation approaches.

Findings

Generated explicit combinatorial formulas for overpartition types.

Connected overpartition theorems with universal mock theta functions.

Enhanced understanding of overpartition identities.

Abstract

In 2015, Bringmann, Lovejoy and Mahlburg considered certain kinds overpartitions, which can been seen as the overpartition analogue of Schur's partition. The motivation of their work is that the difference between the generating function of Schur's classical partitions and the generating functions of the partitions in which the smallest part is excluded. The difference between the two generating functions of partitions is a specialization of the "universal" mock theta function g_3 which introduced by Hickerson. To give an analogue of this, by using another universal mock theta function g_2 instead of g_3, Bringmann Lovejoy and Mahlburg introduced two kinds of overpartitions, which satisfy certain congruence conditions and difference conditions with the smallest parts different. They prove these theorems by using the q-differential equations. In this paper, we will give the generating functions of these two kinds of overpartitions by combinatorial technique.