Desarrangements revisited: statistics and pattern avoidance

TL;DR

This paper revisits desarrangements, a special class of permutations, providing new enumeration formulas based on permutation statistics and exploring pattern avoidance, revealing connections to well-known number sequences.

Contribution

It offers new generating function formulas for counting desarrangements by various statistics and fully enumerates pattern-avoiding desarrangements, linking them to classical number sequences.

Findings

Generated formulas for counting desarrangements by descents, peaks, valleys, double ascents, and double descents.

Enumerated desarrangements avoiding specific length 3 patterns.

Connected pattern-avoiding desarrangements to Catalan, Fine, Jacobsthal, and Fibonacci numbers.

Abstract

A desarrangement is a permutation whose first ascent is even. Desarrangements were introduced in the 1980s by Jacques D\'{e}sarm\'{e}nien, who proved that they are in bijection with derangements. We revisit the study of desarrangements, focusing on two themes: the refined enumeration of desarrangements with respect to permutation statistics, and pattern avoidance in desarrangements. Our main results include generating function formulas for counting desarrangements by the number of descents, peaks, valleys, double ascents, and double descents, as well as a complete enumeration of desarrangements avoiding a prescribed set of length 3 patterns. We find new interpretations of the Catalan, Fine, Jacobsthal, and Fibonacci numbers in terms of pattern-avoiding desarrangements.