The minimal excludant and the least r-gaps of ℓ-regular partitions
Abstract
The minimal excludant (mex) of an integer partition λ is the smallest positive integer that is not a part of λ. In this paper we study an analogue of this statistic for ℓ-regular partitions with ℓ≥2, that is, partitions in which no part is divisible by ℓ. Recently, Meher \cite{Meher} defined the minimal excludant for ℓ-regular partitions. Motivated by Meher \cite{Meher}, for an ℓ-regular partition we define its least r-gap as the smallest positive integer not divisible by ℓ that appears fewer than r times as a part, and we obtain the two-variable generating function for the number of ℓ-regular partitions of n with a given least r-gap. Let σmexℓ(n), and σmexℓ,r(n) be the sum of the of the mex values over all ℓ-regular partitions of n and least r-gaps over all ℓ-regular partitions of n respectively. Using Jacobi's triple product identity, we derive explicit expressions for the generating function of σmexℓ(n), and σmexℓ,r(n), with separate formulas for even and odd ℓ. We also show that the sequence {σmexℓ,r(n)}n≥0 is weakly increasing for every ℓ≥2 and r≥1. Also, using Ingham's Tauberian theorem, we establish Hardy--Ramanujan type asymptotic formulas for σmexℓ,r(n) when ℓ=4,5,7 and r≥1 is arbitrary. Lastly, we prove σmexℓ,r 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.
Related
- The Minimal excludant of ℓ-regular partition
- On the Moments of Least s-Gaps of Partitions and a Conjecture of Baruah and Talukdar
- On the growth of weak regular partitions in k-uniform hypergraphs
- Internal partitions of $5$-regular and $8$-regular graphs
- The integrability of sesqui-regular graphs with smallest eigenvalue at least −3
BibTeX
@misc{priya2026minimal,
title = {{The minimal excludant and the least $r$-gaps of $\ell$-regular partitions}},
author = {Tejash Priya and Sourav Bhowmick and Nabin Kumar Meher},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07871},
eprint = {2611.07871},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07871},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The minimal excludant and the least r-gaps of ℓ-regular partitions
AU - Tejash Priya
AU - Sourav Bhowmick
AU - Nabin Kumar Meher
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07871
UR - https://arxiv.org/abs/2611.07871
AB - The minimal excludant (mex) of an integer partition λ is the smallest positive integer that is not a part of λ. In this paper we study an analogue of this statistic for ℓ-regular partitions with ℓ≥2, that is, partitions in which no part is divisible by ℓ. Recently, Meher \cite{Meher} defined the minimal excludant for ℓ-regular partitions. Motivated by Meher \cite{Meher}, for an ℓ-regular partition we define its least r-gap as the smallest positive integer not divisible by ℓ that appears fewer than r times as a part, and we obtain the two-variable generating function for the number of ℓ-regular partitions of n with a given least r-gap. Let σ^mex_ℓ(n), and σ^mex_ℓ,r(n) be the sum of the of the mex values over all ℓ-regular partitions of n and least r-gaps over all ℓ-regular partitions of n respectively. Using Jacobi's triple product identity, we derive explicit expressions for the generating function of σ^mex_ℓ(n), and σ^mex_ℓ,r(n), with separate formulas for even and odd ℓ. We also show that the sequence {σ^mex_ℓ,r(n)}_n≥0 is weakly increasing for every ℓ≥2 and r≥1. Also, using Ingham's Tauberian theorem, we establish Hardy--Ramanujan type asymptotic formulas for σ^mex_ℓ,r(n) when ℓ=4,5,7 and r≥1 is arbitrary. Lastly, we prove σ^mex_ℓ,r 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 -