Sum rules for permutations with fixed points involving Stirling numbers of the first kind

TL;DR

This paper derives sum rules for permutations with fixed points involving Stirling numbers of the first kind, connecting moments, binomial coefficients, and Bell numbers.

Contribution

It introduces new sum rules for permutations with fixed points using Stirling numbers, expanding combinatorial identities and linking to Bell numbers.

Findings

Derived sum rules involving Stirling numbers of the first kind.

Established connections between permutation moments and binomial coefficients.

Outlined relationships with Bell numbers.

Abstract

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 Schl\"omlich expression of Stirling numbers, we also deduce sum rules for binomial coefficients. Connections with Bell numbers $B_n$ are outlined.