On the Moments of Least s-Gaps of Partitions and a Conjecture of Baruah and Talukdar

Abstract

Andrews and Newman \cite{AN19, AN20} defined the minimal excludant or mex of a partition. The mex of a partition is the smallest positive integer which is absent from the partition. Baruah, Bhoria, Eyyunni and Maji \cite{BBEM} studied the sum of mex split according to parity, together with its ℓ-th moments. Ballantine and Merca \cite{BM20} generalized mex to the least s-gap, which is the smallest natural number that does not occur minimum s times in the partition. Baruah and Talukdar \cite{BT} conjectured a corresponding asymptotic equivalence between the sums of odd and even least s-gaps for every s>1, generalizing a theorem of Barman and Singh \cite{BS} for the classical mex. In this article, we derive exact formulas for the ℓ-th moments of smex(π) for every fixed ℓ≥ 1, in terms of partition functions. We give a complete proof of the conjecture of Baruah and Talukdar \cite{BT} for every natural number s>1. We also generalize an identity of Hopkins, Sellers and Stanton \cite{HSS} to the least s-gap setting.

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BibTeX RIS

BibTeX

@misc{bhowmick2026moments,
  title = {{On the Moments of Least $s$-Gaps of Partitions and a Conjecture of Baruah and Talukdar}},
  author = {Sourav Bhowmick and Nabin Kumar Meher},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.31104},
  eprint = {2609.31104},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31104},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On the Moments of Least s-Gaps of Partitions and a Conjecture of Baruah and Talukdar
AU  - Sourav Bhowmick
AU  - Nabin Kumar Meher
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.31104
UR  - https://arxiv.org/abs/2609.31104
AB  - Andrews and Newman \cite{AN19, AN20} defined the minimal excludant or mex of a partition. The mex of a partition is the smallest positive integer which is absent from the partition. Baruah, Bhoria, Eyyunni and Maji \cite{BBEM} studied the sum of mex split according to parity, together with its ℓ-th moments. Ballantine and Merca \cite{BM20} generalized mex to the least s-gap, which is the smallest natural number that does not occur minimum s times in the partition. Baruah and Talukdar \cite{BT} conjectured a corresponding asymptotic equivalence between the sums of odd and even least s-gaps for every s>1, generalizing a theorem of Barman and Singh \cite{BS} for the classical mex. In this article, we derive exact formulas for the ℓ-th moments of smex(π) for every fixed ℓ≥ 1, in terms of partition functions. We give a complete proof of the conjecture of Baruah and Talukdar \cite{BT} for every natural number s>1. We also generalize an identity of Hopkins, Sellers and Stanton \cite{HSS} to the least s-gap setting.
ER  -

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