Repetition in reduced decompositions

TL;DR

This paper investigates the relationship between repeated letters in reduced decompositions of permutations and pattern occurrences, establishing bounds and exact conditions for equality based on pattern avoidance.

Contribution

It provides a bijective proof linking repeated letters in reduced decompositions to specific pattern avoidance criteria in permutations.

Findings

Number of repeated letters ≤ number of 321- and 3412-patterns in w

Equality holds iff w avoids ten specific patterns

Establishes a pattern avoidance characterization for equality case

Abstract

Given a permutation w, we show that the number of repeated letters in a reduced decomposition of w is always less than or equal to the number of 321- and 3412-patterns appearing in w. Moreover, we prove bijectively that the two quantities are equal if and only if w avoids the ten patterns 4321, 34512, 45123, 35412, 43512, 45132, 45213, 53412, 45312, and 45231.