Reduced Decompositions with One Repetition and Permutation Pattern Avoidance

Abstract

In 2007, Tenner established a connection between pattern avoidance in permutations and the Bruhat order on permutations by showing that the downset of a permutation in the Bruhat order is a Boolean algebra if and only if the permutation is 3412 and 321 avoiding. Tenner conjectured, but did not prove, that if the permutation is 321 avoiding and contains exactly one 3412 pattern, or if the permutation is 3412 avoiding and contains exactly one 321 pattern, then there exists a reduced decomposition with precisely one repetition. This property actually characterizes permutations with precisely one repetition. The goal of this paper is to prove this equivalence as a first step in our program to count permutations with few repetitions of 321 and 3412 and to understand Bruhat downsets by means of pattern avoidance.