Sum rules for permutations with fixed points involving Stirling numbers of the first kind
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918378290020352 |
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| author | Pain, Jean-Christophe |
| author_facet | Pain, Jean-Christophe |
| contents | We propose sum rules for permutations $p_n(k)$ of the ensemble $\left\{1,2,\cdots,n\right\}$ with $k$ fixed points, in the form of partial sums of their moments. The corresponding identities involve Stirling numbers of the first kind $s(q,r)$. Using a formula due to Vassilev-Missana and the Schlömlich expression of Stirling numbers, we also deduce sum rules for binomial coefficients. Connections with Bell numbers $B_n$ are outlined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07284 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sum rules for permutations with fixed points involving Stirling numbers of the first kind Pain, Jean-Christophe Combinatorics We propose sum rules for permutations $p_n(k)$ of the ensemble $\left\{1,2,\cdots,n\right\}$ with $k$ fixed points, in the form of partial sums of their moments. The corresponding identities involve Stirling numbers of the first kind $s(q,r)$. Using a formula due to Vassilev-Missana and the Schlömlich expression of Stirling numbers, we also deduce sum rules for binomial coefficients. Connections with Bell numbers $B_n$ are outlined. |
| title | Sum rules for permutations with fixed points involving Stirling numbers of the first kind |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.07284 |