Simple permutations poset

TL;DR

This paper investigates the structure of simple permutations within the poset defined by pattern involvement, providing a chain characterization that leads to an efficient algorithm for enumerating simple permutations in certain classes.

Contribution

It establishes a chain-based characterization of simple permutations in the poset, enabling a polynomial-time algorithm for their enumeration in wreath-closed classes.

Findings

Chains of simple permutations can be constructed with steps of size 1 or 2.

The characterization leads to a polynomial-time algorithm for enumeration.

Results extend previous work on critically indecomposable posets.

Abstract

This article studies the poset of simple permutations with respect to the pattern involvement. We specify results on critically indecomposable posets obtained by Schmerl and Trotter to simple permutations and prove that if $\sigma, \pi$ are two simple permutations such that $\pi < \sigma$ then there exists a chain of simple permutations $\sigma^{(0)} = \sigma, \sigma^{(1)}, ..., \sigma^{(k)}=\pi$ such that $|\sigma^{(i)}| - |\sigma^{(i+1)}| = 1$ - or 2 when permutations are exceptional- and $\sigma^{(i+1)} < \sigma^{(i)}$. This characterization induces an algorithm polynomial in the size of the output to compute the simple permutations in a wreath-closed permutation class.