Two Descent Statistics over 321-Avoiding Centrosymmetric Involutions

Abstract

Centrosymmetric involutions in the symmetric group ${\mathcal S}_{2n}$ 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 betweencentrosymmetric involutions in ${\mathcal S}_{2n}$ and subsets of $\{1,\dots,n\}$, 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.