Combinatorial proofs and generalization of Bringmann, Lovejoy and Mahlburg’s overpartition theorems

Abstract

In 2013, Bringmann and Mahlburg defined a new type of partitions by adding a different restriction on the smallest parts in Gleissberg’s generalization of partitions considered by Schur. The generating function of this new Schur-type partitions is a mixed mock modular form, more precisely, it equals the product of the generating function of Gleissberg’s generalization and a specialization of a universal mock theta function $g_3(x;q)$, where $g_3(x;q)$ is Hickerson’s universal mock theta function of odd order. Gordon and McIntosh also found a second universal mock theta function of even order $g_2(x;q)$. To give an analogue of Bringmann and Mahlburg’s result of these two Schur-type identities, with respect to $g_2(x;q)$, Bringmann, Lovejoy and Mahlburg investigated two kinds of overpartitions and obtained two overpartition theorems which can be viewed as the overpartition analogues of Gleissberg’s generalization and Bringmann and Mahlburg’s theorem. The quotient of the generating functions of these two kinds of overpartitions is a specialization of $g_2(x;q)$. At the end of their article, Bringmann et al. asked for a bijective proof of their first theorem. In this paper, we construct a bijection by employing the d-modular Ferrers diagram to prove their first theorem and also derive a new bi-parameter generating function for the second set of overpartitions. By this new generating function and a $_3\phi _2$ transformation, we rediscover their second identity and a corollary. Inspired by a generalization of Schur’s theorem due to Andrews, we also give a generalization of Bringmann et al.’s first theorem which extends r, $d-r$ to an integer set A and the modulus from 2d to td.