关于分拆的最小 $r$-间隙矩及 Baruah 与 Talukdar 猜想
On the Moments of Least $r$-Gaps of Partitions and a Conjecture of Baruah and Talukdar
- National Institute of Technology, Raipur（印度理工学院赖布尔分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究分拆的最小 $r$-间隙的 $k$ 阶矩，给出精确公式，并证明 Baruah 与 Talukdar 关于奇偶最小 $r$-间隙和渐近等价的猜想，同时推广了相关恒等式。
AI中文摘要：
最小缺项（mex）由 Andrews 和 Newman 引入，是分拆中缺失的最小正整数。Baruah、Bhoria、Eyyunni 和 Maji 研究了按奇偶性划分的 mex 之和及其 $k$ 阶矩。Ballantine 和 Merca 将 mex 推广为最小 $r$-间隙，即分拆中出现次数少于 $r$ 次的最小自然数。Baruah 和 Talukdar 猜想：对于每个 $r>1$，奇数与偶数最小 $r$-间隙之和之间存在相应的渐近等价关系，这推广了 Barman 和 Singh 关于经典 mex 的定理。本文中，我们针对每个固定的 $k\geq1$，用分拆函数导出了 $r\text{-}\mathrm{mex}(\pi)$ 的 $k$ 阶矩的精确公式。我们完整证明了 Baruah 和 Talukdar 对每个自然数 $r>1$ 的猜想。我们还将 Hopkins、Sellers 和 Stanton 的一个恒等式推广到最小 $r$-间隙的情形。
英文摘要：
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.
