On the number of missing integers in partitions

TL;DR

This paper investigates the distribution and properties of missing positive integers in partitions and overpartitions, including their counts, congruences, and conjectured inequalities.

Contribution

It introduces new results on the number of missing integers in partitions, explores related congruences, and proposes bias inequality conjectures.

Findings

Count of partitions with a fixed number of missing integers

Congruences for functions related to missing integers

Three bias inequality conjectures for these functions

Abstract

In the preceding decade, Andrews and Newman resurrected the concept of a `minimal excludant' of a partition ($mex$, for short), namely, the least positive missing integer in a partition. Subsequently, several authors have not only studied its generalizations, analogues and the like but also connected the mex to several important partition statistics. In the present paper, we study the set of missing positive integers as a whole, in two different classes of partitions, namely, unrestricted partitions and overpartitions. To be precise, a $missing \ integer$ is a positive integer that is less than the largest part of a partition and which does not occur as a part. In particular, we examine the number of partitions with a given number of missing integers, determine congruences for two pairs of functions associated to them, and propose three bias type inequality conjectures for these functions.