A simple bijective proof of a familiar derangement recurrence

TL;DR

This paper presents a straightforward bijective proof of the well-known derangement recurrence relation, offering a simpler combinatorial explanation compared to previous proofs.

Contribution

It introduces a new, simpler bijective proof of the derangement recurrence relation, enhancing understanding of derangement numbers.

Findings

Provides a simpler combinatorial proof of the derangement recurrence

Clarifies the bijective relationship underlying derangement numbers

Contributes to combinatorial proof techniques

Abstract

It is well known that the derangement numbers $d_n$, which count permutations of length $n$ with no fixed points, satisfy the recurrence $d_n=nd_{n-1}+(-1)^n$ for $n\ge1$. Combinatorial proofs of this formula have been given by Remmel, Wilf, D\'esarm\'enien and Benjamin--Ornstein. Here we present yet another, arguably simpler, bijective proof.