Fibonacci Numbers, Reduced Decompositions and 321/3412 Pattern Classes

We provide a bijection from the permutations in Sn that avoid 3412 and contain exactly one 321 pattern to the permutations in Sn+1 that avoid 321 and contain exactly one 3412 pattern. The enumeration of these classes is obtained from their classification via reduced decompositions. The results are extended to involutions in the above pattern classes using reduced decompositions reproducing a result of Egge. 1. Permutation Patterns and Reduced Decompositions Throughout this paper, permutations will be written in one-line notation, π = π1 . . . πn, where the image of i under π is πi. Definition 1.1. A permutation π = π1π2 . . . πn ∈ Sn is said to contain a permutation σ = σ1 . . . σm ∈ Sm if there exists a subsequence 1 ≤ i1 < i2 < . . . im ≤ n such that σj < σk if and only if πij < πik . If π does not contain σ, then π avoids σ. Let π ∈ Sm. We denote the set of permutations in Sn that avoid π by Avn(π). Definition 1.2. A reduced decomposition of π ∈ Sn is a word s1 . . . sk where each sj is a transposition of the form (i, i+ 1) for some i such that π = s1 . . . sk and k is as small as possible. k is the length of the permutation denoted l(π). Reduced decompositions are not unique. For example, (12)(23)(12) = 321 = (23)(12)(23). In order to simplify the notation, we will write i to represent the transposition (i, i + 1) and to distinguish expressions on the transpositions (i, i + 1) (particularly reduced decompositions) from permutations, we will put brackets around such expressions. For instance, [121] = (12)(23)(12) = 321. Definition 1.3. A factor in a reduced decomposition [i1 . . . ik] is a consecutive substring. [321323] is a reduced decomposition for 4321. [213] is a factor, but [313] is not. Any reduced decomposition for a fixed permutation π can be transformed into any other by the use of braid moves. The two braid moves are: • (Short Braid Move) [ij] = [ji] if |i− j| > 1. • (Long Braid Move) [i(i+ 1)i] = [(i+ 1)i(i+ 1)] for all i. Reduced decompositions have some very special properties. The following holds more generally for any reduced decomposition in any Coxeter group, but we will only need it for reduced decompositions of permutations. The proof of this property can be found in [5] or [2]. Theorem 1.4. (Exchange Property) Let [s1 . . . sk] be a reduced decomposition for π ∈ Sn and let s be any transposition of the form (i, i + 1). If l([s1 . . . sks]) < l([s1 . . . sks]), then [s1 . . . sk] = [s1 . . . ŝj . . . sk] for some 1 ≤ j ≤ k.