Identities for permutations with fixed points

TL;DR

The paper addresses fixed-point statistics in permutations by linking them to the determinant of a structured matrix $M_x$ whose expansion encodes fixed-point counts. It derives two families of sum rules: (i) by successive differentiation of $\det M_x$, giving explicit identities for sums weighted by falling factorials in the fixed-point statistic, and (ii) by successive integration, yielding identities for sums of $\frac{(\mathrm{fix}(\sigma))!}{(\mathrm{fix}(\sigma)+k)!}$. The main results include closed-form expressions such as $\sum_{\sigma\in\mathfrak{S}_n}\epsilon(\sigma)\frac{(\mathrm{fix}(\sigma))!}{(\mathrm{fix}(\sigma)+k)!}=(-1)^{n+1}\frac{n^2+(k-1)n-(k-1)}{(k-1)!(n+k-1)(n+k)}$ (for $k\ge2$) and the derivative-based identities for $k$-fold fixed-point products, with special cases at $x=1$ and $x=2$ explicitly described. The work connects fixed-point statistics to determinant calculus and suggests further sum rules derived from the same matrix approach.

Abstract

We present identities for permutations with fixed points. The formulas are based on successive derivations or integrations of the determinant of a particular matrix.

Theorems & Definitions (5)

• Definition 1
• Theorem 1
• proof
• Theorem 2
• proof