Generalized mex-functions and partitions with congruence restrictions on parts

In a recent work, Andrews and Newman introduced the statistic [Formula: see text] of a partition [Formula: see text], and obtained many interesting identities concerning the functions [Formula: see text] and [Formula: see text], the number of partitions of n whose [Formula: see text] are congruent to a and [Formula: see text] modulo [Formula: see text], respectively. Their pioneering work attracted much attention and inspired numerous subsequent studies. In this paper, motivated by the idea of Andrews–Newman, we consider a more general counting function [Formula: see text] induced by [Formula: see text], the number of partitions of n whose [Formula: see text] are congruent to [Formula: see text] modulo [Formula: see text] for [Formula: see text], and obtain a series of equality relations that connect the difference of two distinct [Formula: see text] functions to the number of partitions with congruence restrictions on parts. Moreover, we extend the counting functions [Formula: see text] to overpartitions as [Formula: see text], and then obtain some relevant equalities.