Sorting and generating reduced words

TL;DR

The paper introduces a new partial order called the directed-braid poset on reduced words of permutations, enabling algorithms for sorting to the natural word and generating all reduced words.

Contribution

It presents a novel partial order structure on reduced words and develops algorithms for sorting and generating these words based on this structure.

Findings

Algorithms successfully produce the natural reduced word.

The generation algorithm enumerates all reduced words of a permutation.

The directed-braid poset structure facilitates these processes.

Abstract

We introduce a partial order on the set of all reduced words of a given permutation $\omega$, called \emph{directed-braid poset} of $\omega$. This poset enables us to produce two algorithms: One is a sorting algorithm applied on any reduced word of $\omega$ and aims to obtained the natural word (lexicographically largest reduced word); the other one is a generation algorithm applied on the natural word and aims to obtained the set of all reduced words of $\omega$.