Reduced Decompositions with One Repetition and Permutation Pattern Avoidance

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 mentioned, 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 understand Bruhat downsets by means of pattern avoidance. 1. Notation, Basic Definitions and Previous Results We will denote the set of all permutations of {1, . . . , n} by Sn and we will write an element π ∈ Sn in one-line notation as π = π1π2 . . . πn, where the image of i under π is πi. So, in the permutation 3214, π(1) = 3, π(2) = 2, etc. We compose permutations from left to right. Definition 1.1. We say a permutation π = π1π2 . . . πn ∈ Sn contains a permutation σ = σ1 . . . σm ∈ Sm as a pattern if there exist i1 < i2 < · · · < im such that πij < πik if and only if σj < σk. If π does not contain σ then we say π avoids σ. Example 1.2. The permutation 7651324 contains the pattern 321, but avoids 231. Permutation patterns are well-studied and we refer the interested reader to [5] or [2] for more information. We will now briefly review some of the results on reduced decompositions that are important here. Recall that Sn is generated by the transpositions (i, i+ 1) for 1 ≤ i < n. Definition 1.3. A reduced decomposition of π ∈ Sn is a word j1 . . . jl where each jk is a transposition of the form (i, i+ 1) such that π = j1 . . . jl and l is as small as possible. Example 1.4. One reduced decomposition of 43152 is (34)(23)(12)(34)(23)(45) We will, from now on, condense the notation of reduced decompositions and just write the first element i of the transposition (i, i+ 1) in reduced decompositions. In the previous example, the reduced decomposition will be condensed to [321324]. We put brackets around the reduced decompositions to distinguish them from the actual permutations. Note, reduced decompositions of permutations are not unique and it is a well known result [4] that any reduced decomposition of π may be obtained from any other by the use of the two braid moves: [ij] = [ji] if |i− j| > 1 [i(i+ 1)i] = [(i+ 1)i(i+ 1)] for all i We will denote by R(π) the set of all reduced decompositions of π and use bold lower-case letters to refer to an element of R(π). Ex: j ∈ R(π), j = [j1 . . . jl]. The number of letters in any reduced decomposition is called the length of the permutation and corresponds to the number of inversions, or occurrences of 21 patterns, in the permutation. We use l(π) to denote the length of the permutation π.