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

Source abstract

The minimal excludant or mex of a partition, introduced by Andrews and Newman \cite{AN2019,AN2020}, is the smallest positive integer missing from that partition. Baruah, Bhoria, Eyyunni and Maji \cite{BBEM2023} studied the sum of mex split according to parity, together with its $k$-th moments. Ballantine and Merca \cite{BM2020} generalized mex to the least $r$-gap, which is the smallest natural number that does not appear at least $r$ times in the partition. Baruah and Talukdar \cite{BT2026} conjectured a corresponding asymptotic equivalence between the sums of odd and even least $r$-gaps for every $r>1$, generalizing a theorem of Barman and Singh \cite{BS2024} for the classical mex. In this article, we derive exact formulas for the $k$-th moments of $r\text{-}\mathrm{mex}(π)$ for every fixed $k\geq1$, in terms of partition functions. We give a complete proof of the conjecture of Baruah and Talukdar \cite{BT2026} for every natural number $r>1$. We also generalize an identity of Hopkins, Sellers and Stanton to the least $r$-gap setting.