The largest and the smallest fixed points of permutations

TL;DR

This paper offers new combinatorial interpretations of derangement numbers by relating them to fixed points of permutations, along with proofs and a recurrence relation.

Contribution

It introduces novel interpretations of derangement numbers based on fixed points and provides new proofs and a recurrence relation.

Findings

Derangement numbers equal the sum of largest fixed points of non-derangements.

Sum of smallest fixed points corresponds to permutations with at least two fixed points.

Provides analytic and bijective proofs for these interpretations.

Abstract

We give a new interpretation of the derangement numbers d_n as the sum of the values of the largest fixed points of all non-derangements of length n-1. We also show that the analogous sum for the smallest fixed points equals the number of permutations of length n with at least two fixed points. We provide analytic and bijective proofs of both results, as well as a new recurrence for the derangement numbers.