Deciding the finiteness of the number of simple permutations contained in a wreath-closed class is polynomial

TL;DR

This paper introduces an efficient O(n ln n) algorithm to determine whether a wreath-closed permutation class contains finitely many simple permutations, improving upon previous high-complexity decision procedures.

Contribution

The paper presents a novel polynomial-time algorithm for deciding finiteness of simple permutations in wreath-closed classes, leveraging automata theory and previous combinatorial results.

Findings

Algorithm runs in O(n ln n) time

Decides finiteness of simple permutations in wreath-closed classes

Transforms the problem into a co-finiteness automaton problem

Abstract

We present an algorithm running in time O(n ln n) which decides if a wreath-closed permutation class Av(B) given by its finite basis B contains a finite number of simple permutations. The method we use is based on an article of Brignall, Ruskuc and Vatter which presents a decision procedure (of high complexity) for solving this question, without the assumption that Av(B) is wreath-closed. Using combinatorial, algorithmic and language theoretic arguments together with one of our previous results on pin-permutations, we are able to transform the problem into a co-finiteness problem in a complete deterministic automaton.