Bijections for restricted inversion sequences and permutations with fixed points

TL;DR

This paper establishes bijective proofs connecting restricted inversion sequences without three consecutive equal entries to permutations with fixed points, offering new combinatorial insights and recurrences.

Contribution

It provides the first bijective proofs linking specific inversion sequences to permutations with fixed points and introduces new recurrence relations for non-derangements.

Findings

Established a bijective proof relating inversion sequences and permutations with fixed points.

Derived simple recurrences for counting non-derangements.

Connected combinatorial structures through explicit bijections.

Abstract

We provide a bijective proof of a formula of Auli and the author expressing the number of inversion sequences with no three consecutive equal entries in terms of the number of non-derangements, that is, permutations with fixed points. Additionally, we give bijective proofs of two simple recurrences for the number of non-derangements.