Descent sets on 321-avoiding involutions and hook decompositions of partitions

Abstract

We show that the distribution of the major index over the set of involutions in S n that avoid the pattern 321 is given by the q-analogue of the n-th central binomial coefficient. The proof consists of a composition of three non-trivial bijections, one being the Robinson-Schensted correspondence, ultimately mapping those involutions with major index m into partitions of m whose Young diagram fits inside a ⌊ n 2 ⌋ × ⌈ n 2 ⌉ box. We also obtain a refinement that keeps track of the descent set, and we deduce an analogous result for the comajor index of 123-avoiding involutions.

Two refined major-balance identities on 321-avoiding involutions

“…Recently, Barnabei et al [2] proved that the distribution of the major index over 321-avoiding involutions is given by…”

“…To prove Theorem 1.2, we make use of a bijection given by Barnabei et al [2] to transform a partial Dyck path of length n into a lattice path going from (0, 0) to (⌈ n 2 ⌉, ⌊ n 2 ⌋) using N and E steps without restriction, called a grand Dyck path of length n. It is known that the area above the grand Dyck path within the ⌊ n 2 ⌋ × ⌈ n 2 ⌉ rectangle coincides with the major index of 321-avoiding involutions. Then we prove Theorem 1.2 by establishing an area-parity-reversing involution on grand Dyck paths (see Propositions 5.4, 5.8 and 5.11).…”

“…In Section 3 we establish an ℓ-parity-reversing involutions on cluster 2-trees, which leads to a proof of Theorem 1.1. In Section 4 we describe a bijection that links partial Dyck paths to grand Dyck paths given in [2]. In Section 5 we establish an area-parity-reversing involution on grand Dyck paths, which leads to a proof of Theorem 1.2.…”

“…There is an alternative construction, given by Barnabei et al [2], for the map δ. First, apply the Robinson-Schensted algorithm to associate an involution σ ∈ I n (321) with a pair of identical n-cell standard Young tableaux with at most two rows.…”

“…In this section we describe a sequence of bijections that link 321-avoiding involutions to grand Dyck paths given by Barnabei et al [2].…”