Sum of elements preceding records in set partitions
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909830926565376 |
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| author | Asakly, Walaa Kezil, Noor |
| author_facet | Asakly, Walaa Kezil, Noor |
| contents | 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 $\sumelements$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06521 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sum of elements preceding records in set partitions Asakly, Walaa Kezil, Noor Combinatorics 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 $\sumelements$. |
| title | Sum of elements preceding records in set partitions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.06521 |