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

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 an n/2 by 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

Families of major index distributions: Closed forms and unimodality

Closed forms for fλ,i(q):=∑τ∈SYT(λ):des(τ)=iqmaj(τ)fλ,i(q):=∑τ∈SYT(λ):des(τ)=iqmaj(τ), the distribution of the major index over standard Young tableaux of given shapes and specified number of descents, are established for a large collection of λλ and ii. Of particular interest is the family that gives a positive answer to a question of Sagan and collaborators. All formulas established in the paper are unimodal, most by a result of Kirillov and Reshetikhin. Many can be identified as specializations of Schur functions via the Jacobi-Trudi identities. If the number of arguments is sufficiently large, it is shown that any finite principal specialization of any Schur function sλ(1,q,q2,…,qn−1)sλ(1,q,q2,…,qn−1) has a combinatorial realization as the distribution of the major index over a given set of tableaux

Geometric and Topological Combinatorics

The 2007 Oberwolfach meeting “Geometric and Topological Combinatorics” presented a great variety of investigations where topological and algebraic methods are brought into play to solve combinatorial and geometric problems, but also where geometric and combinatorial ideas are applied to topological questions

Two descent statistics over 321-avoiding centrosymmetric involutions

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 distribu- tion of some natural descent statistics on 321-avoiding centrosymmetric involutions, including the number of descents in the rst 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 rene 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