This article explores the properties of the minimal excludant function on integer partitions, as introduced by G. E. Andrews and D. Newman [Ann. Comb. 23, No. 2, 249–254 (2019; Zbl 1458.11011)]. Given a partition \(\pi\) of a positive integer \(n\), the minimal excludant, denoted \(\mathrm{mex}(\pi)\), is the smallest positive integer that is not a part of \(\pi\).
The paper proves several congruence and multiplicative formulas for \(\sigma\mathrm{mex}(n)\) – which is defined as the sum of \(\mathrm{mex}(\pi)\) taken over all partitions \(\pi\) of \(n\), showing that it can be expressed in terms of certain eta quotients, modular forms, and Hecke eigenforms. The author also considers congruences of \(\mathrm{moex}(\pi)\) – the smallest odd integer that is not a part of \(\pi\), and properties of \(\sigma\mathrm{moex}(n)\) – the sum of \(\mathrm{moex}(\pi)\) taken over all partitions \(\pi\) of \(n\).
The main result of the paper concerns the distribution of the values of \(\sigma\mathrm{moex}(n)\) for positive integers \(n\). The authors show that for any sufficiently large \(X\), the number of positive integers \(n \leq X\) such that \(\sigma\mathrm{moex}(n)\) is even (or odd) is at least \(\mathrm{O}(\log \log X)\). This result is proved using techniques from analytic number theory and modular forms.
Overall, this paper provides a detailed analysis of the minimal excludant function on integer partitions and its related functions, as well as new results on their divisibility and distribution. The paper may be of interest to researchers in combinatorics, number theory, and modular forms.