A Statistic on Involutions

We define a statistic, called weight, on involutions and consider two applications in which this statistic arises. Let I(n) denote the set of all involutions on n](={1,2,..., n}) and let F(2n) denote the set of all fixed point free involutions on 2n]. For an involution , let || denote the number of 2-cycles in . Let n] _q =1+q++q^n-1 and let denote the q-binomial coefficient. There is a statistic wt on I(n) such that the following results are true.(i) We have the expansion (ii) An analog of the (strong) Bruhat order on permutations is defined on F(2n) and it is shown that this gives a rank-2 graded EL-shellable poset whose order complex triangulates a ball. The rank of F(2n) is given by wt() and the rank generating function is 1] _q 3]q2n-1]q.