Two characterizations of the shape of the base poset derived from the Lehmer code of a permutation using permutation patterns

摘要

The Lehmer code is a classical and fundamental permutation code which encodes information about the inversions of a permutation. Denoncourt constructed a poset M_ω which is the subposet of joinirreducible elements of the Lehmer codes of the permutations in [e_n, ω] in the left weak Bruhat order, i.e., the inversion order, on S_n for ω ? S_n. In this paper, we investigate the poset structure of M_ω in terms of pattern avoidance. First we show that M_ω is a B_2-free poset if and only if ω is a 3412-3421-avoiding permutation. Next we prove that Mw is poset isomorphic to the corresponding root poset if and only if ω is a 321-avoiding permutation. AMS 2000 SUBJECT CLASSIFICATIONS:06A07, 05A05.