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

Asymptotic distribution of fixed points of pattern-avoiding involutions

For a variety of pattern-avoiding classes, we describe the limiting distribution for the number of fixed points for involutions chosen uniformly at random from that class. In particular we consider monotone patterns of arbitrary length as well as all patterns of length 3. For monotone patterns we utilize the connection with standard Young tableaux with at most $k$ rows and involutions avoiding a monotone pattern of length $k$. For every pattern of length 3 we give the bivariate generating function with respect to fixed points for the involutions that avoid that pattern, and where applicable apply tools from analytic combinatorics to extract information about the limiting distribution from the generating function. Many well-known distributions appear.Comment: 16 page