Reduced word manipulation: patterns and enumeration

TL;DR

This paper introduces reduced word manipulation techniques to analyze permutations, revealing new connections between pattern containment, reduced words, and enumeration, with applications to pattern avoidance and combinatorial structures.

Contribution

It establishes a broad link between pattern containment and reduced words, and explores their enumeration and structural properties, extending previous work on vexillary permutations.

Findings

Reduced words enumeration increases with pattern containment.

Permutations and patterns have equal reduced words if they share the same length.

A bijection exists between 132-avoiding permutations and partitions.

Abstract

We develop the technique of reduced word manipulation to give a range of results concerning reduced words and permutations more generally. We prove a broad connection between pattern containment and reduced words, which specializes to our previous work for vexillary permutations. We also analyze general tilings of Elnitsky's polygon, and demonstrate that these are closely related to the patterns in a permutation. Building on previous work for commutation classes, we show that reduced word enumeration is monotonically increasing with respect to pattern containment. Finally, we give several applications of this work. We show that a permutation and a pattern have equally many reduced words if and only if they have the same length (equivalently, the same number of 21-patterns), and that they have equally many commutation classes if and only if they have the same number of 321-patterns. We also apply our techniques to enumeration problems of pattern avoidance, and give a bijection between 132-avoiding permutations of a given length and partitions of that same size, as well as refinements of this data and a connection to the Catalan numbers.