Congruences for some partitions related to mock theta functions

摘要

Partitions related to mock theta functions were widely studied in the literature. Recently, Andrews et al. introduced two new kinds of partitions counted by p(omega)(n) and p(nu)(n), whose generating functions are q omega(q) and nu(-q), where omega(q) and nu(q) are two third mock theta functions. Meanwhile, they obtained some congruences for p(omega)(n), p(nu)(n), and the associated smallest parts function spt(omega)(n). Furthermore, Andrews et al. discussed the associated smallest parts function spt(omega)(n). Furthermore, Andrews et al. discussed the overpartition analogues of p(omega)(n) and spt(omega)(n) which are denoted by (p)over-bar(omega)(n) and (spt)over-bar(omega)(n). In this paper, we derive more congruences for p(omega)(n), p(nu)(n), (p)over-bar(omega)(n), (spt)over-bar(omega)(n), and (spt)over-bar(n). Moreover, we establish some congruences for (p)over-bar(sigma)(n) and its associated smallest parts function (spt)over-bar(sigma)(n), where (p)over-bar(sigma)(n) denotes the number of overpartitions of n such that all even parts are at most twice the smallest part, and in which the smallest part is always overlined.