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 a ⌊ n 2 ⌋ × ⌈ 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.