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arXiv · 2510.06521

Sum of elements preceding records in set partitions

Walaa Asakly·Noor Kezil
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Abstract

In this paper, we aim to derive an explicit formula for the total number of elements preceding records over all set partitions of $[n]$ with exactly $k$ blocks, as well as an asymptotic estimate for the total sum of elements preceding records in all set partitions of $[n]$, expressed in terms of Bell numbers. To achieve this, we analyze the generating function that enumerates set partitions of $[n]$ according to this statistic, which we denote by $\sumelements$.

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Walaa Asakly, Noor Kezil. 2025-10-07. Sum of elements preceding records in set partitions. https://arxiv.org/abs/2510.06521

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