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.

2013年，Bringmann和Mahlburg通过在Schle认为的Gleissberg分区一般化中对最小部分添加不同的限制，定义了一种新型分区。这种新的Schur型分区的生成函数是混合的模拟模块化形式，更确切地说，它等于Gleissberg泛化的生成函数与通用模拟theta函数\（g_3（x; q）\）的乘积的乘积。，其中\（g_3（x; q）\）是Hickerson的奇数阶通用模拟theta函数。戈登和麦金托什还发现了第二个偶数\（g_2（x; q）\）的通用模拟theta函数。给出关于\（g_2（x; q）\）这两个Schur型恒等式的Bringmann和Mahlburg结果的类似物，Bringmann，Lovejoy和Mahlburg研究了两种超分区，并获得了两个超分区定理，这些定理可以看作是Gleissberg泛化以及Bringmann和Mahlburg定理的超分区类似物。这两种超分区的生成函数的商是\（g_2（x; q）\）的特化。在文章末尾，Bringmann等人。要求他们的第一个定理的双射证明。在本文中，我们通过使用d模Ferrers图构造一个双射图以证明其第一个定理，并为第二组超划分导出一个新的双参数生成函数。通过这个新的生成函数和一个\（_ 3 \ phi _2 \）转变，我们重新发现了他们的第二个身份和必然结果。受安德鲁斯（Andrews）对舒尔定理的推广的启发，我们还给出了Bringmann等人的第一个定理的推广，该定理将r，\（dr \）扩展为整数集A，模数从2 d扩展到td。