A Short Combinatorial Proof of Derangement Identity

TL;DR

This paper provides a concise combinatorial proof for derangement identities involving rencontres numbers, which count permutations with a specified number of fixed points, including derangements with no fixed points.

Contribution

It introduces a novel, short combinatorial proof for weighted sum derangement identities, enhancing understanding of permutation fixed point distributions.

Findings

Simplifies proof of derangement identities

Clarifies relationship between rencontres numbers and derangements

Provides a new combinatorial approach

Abstract

The $n$-th rencontres number with the parameter $r$ is the number of permutations having exactly $r$ fixed points. In particular, a derangement is a permutation without any fixed point. We presents a short combinatorial proof for a weighted sum derangement identities.