A relation between the shape of a permutation and the shape of the base poset derived from the Lehmer codes

TL;DR

This paper explores the relationship between permutation shapes and the structure of associated posets derived from Lehmer codes, revealing a characterization of certain permutation classes via poset properties.

Contribution

It establishes a precise equivalence between 3412-3421-avoiding permutations and the $B_2$-free property of the related posets, linking permutation patterns to poset structure.

Findings

$M_{\omega}$ is $B_2$-free iff $\omega$ avoids 3412 and 3421 patterns

Characterization of permutation classes via poset properties

Connection between permutation pattern avoidance and poset structure

Abstract

For a permutation $\omega \in S_{n}$ Denoncourt constructed a poset $M_{\omega}$ which is the set of join-irreducibles of the Lehmer codes of the permutations in $[e, \omega]$ in the inversion order on $S_{n}$. In this paper we show that $M_{\omega}$ is a $B_{2}$-free poset if and only if $\omega$ is a 3412-3421-avoiding permutation.