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 δ, let |δ| denote the number of 2-cycles in δ. Let [n]q = 1+q+· · ·+qn−1 and let (k)q 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 ( n k ) q = ∑ δ∈I (n) (q − 1)|δ|qwt(δ) ( n − 2|δ| k − |δ| )