Permutree sorting

TL;DR

The paper introduces the permutree sorting algorithm, a generalization of existing permutation sorting methods, which characterizes permutations based on specific subword patterns and explores related automata for sorting validation.

Contribution

It defines the permutree sorting algorithm, extending stack and c-sorting, and analyzes the automaton structure for permutation sorting based on pattern avoidance.

Findings

Characterizes permutations rejected by the permutree sorting algorithm.

Establishes automaton acceptance criteria for permutree sortable permutations.

Generalizes existing sorting algorithms to a broader permutation class.

Abstract

Generalizing stack sorting and $c$-sorting for permutations, we define the permutree sorting algorithm. Given two disjoint subsets $U$ and $D$ of $\{2, \dots, n-1\}$, the $(U,D)$-permutree sorting tries to sort the permutation $\pi \in \mathfrak{S}_n$ and fails if and only if there are $1 \le i < j < k \le n$ such that $\pi$ contains the subword $jki$ if $j \in U$ and $kij$ if $j \in D$. This algorithm is seen as a way to explore an automaton which either rejects all reduced expressions of $\pi$, or accepts those reduced expressions for $\pi$ whose prefixes are all $(U,D)$-permutree sortable.