Divisibility and distribution of mex-related integer partitions of Andrews and Newman

Abstract

Andrews and Newman introduced the minimal excludant or “[Formula: see text]” function for an integer partition [Formula: see text] of a positive integer [Formula: see text], [Formula: see text], as the smallest positive integer that is not a part of [Formula: see text]. They defined [Formula: see text] to be the sum of [Formula: see text] taken over all partitions [Formula: see text] of [Formula: see text]. We prove infinite families of congruence and multiplicative formulas for [Formula: see text]. By restricting to the part of [Formula: see text], Andrews and Newman also introduced [Formula: see text] to be the smallest odd integer that is not a part of [Formula: see text] and [Formula: see text] to be the sum of [Formula: see text] taken over all partitions [Formula: see text] of [Formula: see text]. In this paper, we show that for any sufficiently large [Formula: see text], the number of all positive integer [Formula: see text] such that [Formula: see text] is an even (or odd) number is at least [Formula: see text].