# Reduced word enumeration, complexity, and randomization

## TL;DR

This paper investigates the complexity and algorithms for enumerating reduced words of permutations, revealing exponential growth in certain statistics and analyzing related combinatorial structures with implications in algebraic geometry.

## Contribution

It provides a formal runtime analysis of the transition algorithm and establishes exponential growth results for the Edelman-Greene statistic, extending to Hecke words and Young tableaux enumeration.

## Key findings

- Edelman-Greene statistic is typically exponentially large
- Transition algorithm's runtime analysis shows exponential complexity
- Enumeration of Hecke words and Young tableaux is also exponential

## Abstract

A reduced word of a permutation $w$ is a minimal length expression of $w$ as a product of simple transpositions. We examine the computational complexity, formulas and (randomized) algorithms for their enumeration. In particular, we prove that the Edelman-Greene statistic, defined by S. Billey-B. Pawlowski, is typically exponentially large. This implies a result of B. Pawlowski, that it has exponentially growing expectation. Our result is established by a formal run-time analysis of A. Lascoux-M.-P. Sch\"utzenberger's transition algorithm. The more general problem of Hecke word enumeration, and its closely related question of counting set-valued standard Young tableaux, is also investigated. The latter enumeration problem is further motivated by work on Brill-Noether varieties due to M. Chan-N. Pflueger and D. Anderson-L. Chen-N. Tarasca.