Pattern-functions, statistics, and shallow permutations

TL;DR

This paper explores the relationship between permutation statistics and pattern-functions, providing new characterizations of shallow permutations and their subclasses using various pattern types and enumerations.

Contribution

It introduces a novel framework linking permutation statistics with pattern-functions and characterizes shallow permutations through multiple pattern representations.

Findings

Characterization of shallow permutations using vincular, mesh, and arrow patterns.

Enumeration of shallow involutions and cycles involving Motzkin and Schr"oder numbers.

Unified approach to permutation depth and pattern enumeration.

Abstract

We study relationships between permutation statistics and pattern-functions, counting the number of times particular patterns occur in a permutation. This allows us to write several familiar statistics as linear combinations of pattern counts, both in terms of a permutation and in terms of its image under the fundamental bijection. We use these enumerations to resolve the question of characterizing so-called "shallow" permutations, whose depth (equivalently, disarray/displacement) is minimal with respect to length and reflection length. We present this characterization in several ways, including vincular patterns, mesh patterns, and a new object that we call "arrow patterns." Furthermore, we specialize to characterizing and enumerating shallow involutions and shallow cycles, encountering the Motzkin and large Schr\"oder numbers, respectively.