Two descent statistics over \(321\)-avoiding centrosymmetric involutions

Marilena Barnabei¹ ; Flavio Bonetti¹ ; Sergi Elizalde² ; Matteo Silimbani¹

¹Università di Bologna
²Dartmouth College

The electronic journal of combinatorics, Tome 23 (2016) no. 1

Résumé

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.

Détail

DOI : 10.37236/5531

Classification : 05A05, 05A19, 05A17, 05A15, 05A30, 05A10, 05E10
Mots-clés : involution, descent, centrosymmetric, pattern avoidance, major index, fixed point, Young diagram, lattice path, hyperoctahedral group, symmetric matching, nonesting, excedance