“The Minimal excludant of ℓ-regular partition” (2026), by Nabin Kumar Meher. Mathematics research paper. Original abstract below, unchanged. A free PDF is available.

The Minimal excludant of ℓ-regular partition

Nabin Kumar Meher · November 4, 2026 · Mathematics

Abstract

Andrews and Newman cite{andrewsnewmanI} introduced and studied ``mex,'' function for an integer partition λ of a positive integer n. The related function σ^mex(n) was subsequently defined, and its generating function and other properties have been studied by several authors. In this article, for a positive integer ℓ ≥ 2, we define the minimal excludant of an ℓ-regular partition λ of a positive integer n. We then define σ_ℓ^mex(n), the sum of the minimal excludants over all ℓ-regular partitions of n, and obtain explicit generating functions for σ_ℓ^mex(n) in the cases ℓ = 2 and ℓ = 3, expressed as eta-quotients. We prove that the sequence σ_ℓ^mex(n) is weakly increasing for every ℓ, and we obtain Hardy--Ramanujan type asymptotic formulas for σ₂^mex(n) and σ₃^mex(n) as n → ∞. Additionally, we completely determine the parity of σ₂^mex(n), showing that it is odd precisely at generalized pentagonal numbers, and we use this characterization to compute the density of n for which σ₂^mex(n) is odd versus even. Finally, for σ₃^mex(n), we prove that it is odd infinitely often, and we establish a quantitative lower bound, showing that the number of such n up to X grows at least like log log X.

Free PDF DOI: 10.48550/arxiv.2611.06072