The depth of a permutation

TL;DR

This paper introduces the depth statistic for Coxeter group elements, especially in the symmetric group, relating it to permutation patterns and classical combinatorial enumeration, providing new characterizations and insights.

Contribution

It defines the depth statistic in Coxeter groups, characterizes permutations where depth equals length or reflection length, and connects these to pattern avoidance and Catalan number enumeration.

Findings

Depth equals sum_i max{w(i)-i, 0} in symmetric groups.

Permutations with depth equal to length are 321-avoiding, counted by Catalan numbers.

Permutations with depth equal to reflection length avoid 321 and 3412, known as boolean permutations.

Abstract

For the elements of a Coxeter group, we present a statistic called depth, defined in terms of factorizations of the elements into products of reflections. Depth is bounded above by length and below by the average of length and reflection length. In this article, we focus on the case of the symmetric group, where we show that depth is equal to sum_i max{w(i)-i, 0}. We characterize those permutations for which depth equals length: these are the 321-avoiding permutations (and hence are enumerated by the Catalan numbers). We also characterize those permutations for which depth equals reflection length: these are permutations avoiding both 321 and 3412 (also known as boolean permutations, which we can hence also enumerate). In this case, it also happens that length equals reflection length, leading to a new perspective on a result of Edelman.