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

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.