The Minimal excludant of ℓ-regular partition
Abstract
Andrews and Newman \cite{andrewsnewmanI} introduced and studied ``mex,'' function for an integer partition λ of a positive integer n. The related function σmex(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 σℓmex(n), the sum of the minimal excludants over all ℓ-regular partitions of n, and obtain explicit generating functions for σℓmex(n) in the cases ℓ = 2 and ℓ = 3, expressed as eta-quotients. We prove that the sequence σℓmex(n) is weakly increasing for every ℓ, and we obtain Hardy--Ramanujan type asymptotic formulas for σ2mex(n) and σ3mex(n) as n → ∞. Additionally, we completely determine the parity of σ2mex(n), showing that it is odd precisely at generalized pentagonal numbers, and we use this characterization to compute the density of n for which σ2mex(n) is odd versus even. Finally, for σ3mex(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.
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BibTeX
@misc{meher2026minimal,
title = {{The Minimal excludant of $\ell$-regular partition}},
author = {Nabin Kumar Meher},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06072},
eprint = {2611.06072},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06072},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Minimal excludant of ℓ-regular partition
AU - Nabin Kumar Meher
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06072
UR - https://arxiv.org/abs/2611.06072
AB - 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 σ_2^mex(n) and σ_3^mex(n) as n → ∞. Additionally, we completely determine the parity of σ_2^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 σ_2^mex(n) is odd versus even. Finally, for σ_3^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.
ER -