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 $\ell$-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 $\ell$-th moments of $\smex(π)$ for every fixed $\ell \ge 1$, 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.

arXiv comment10 pages