Two descent statistics over 321-avoiding centrosymmetric involutions

TL;DR

This paper studies descent statistics in 321-avoiding centrosymmetric involutions, revealing their distribution through new bijections and refining known combinatorial results related to the major index.

Contribution

It introduces two novel bijections that connect centrosymmetric involutions with subsets and Young diagrams, advancing the understanding of descent distributions.

Findings

Distribution of descent statistics characterized

New bijections established between involutions and combinatorial objects

Refinement of the major index distribution using q-analogues

Abstract

Centrosymmetric involutions in the symmetric group S_{2n} are permutations \pi such that \pi=\pi^{-1} and \pi(i)+\pi(2n+1-i)=2n+1 for all i, and they are in bijection with involutions of the hyperoctahedral group. We describe the distribution of some natural descent statistics on 321-avoiding centrosymmetric involutions, including the number of descents in the first half of the involution, and the sum of the positions of these descents. Our results are based on two new bijections, one between centrosymmetric involutions in S_{2n} and subsets of {1,...,n}, and another one showing that certain statistics on Young diagrams that fit inside a rectangle are equidistributed. We also use the latter bijection to refine a known result stating that the distribution of the major index on 321-avoiding involutions is given by the q-analogue of the central binomial coefficients.