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

The joint distribution of consecutive patterns and descents in permutations avoiding 3-1-2

We exploit Krattenthaler’s bijection between the set Sn(3-1-2) of permutations in Sn avoiding the classical pattern 3-1-2 and Dyck n-paths to study the joint distribution over the set Sn(3-1-2) of a given consecutive pattern of length 3 and of descents. We utilize a involution on Dyck paths due to E. Deutsch to show that these consecutive patterns split into 3 equidistribution classes. In addition, we state equidistribution theorems concerning quadruplets of statistics relative to occurrences of consecutive patterns of length 3 and of descents in a permutation