A statistic on involutions

Abstract

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 delta, let \ delta \ denote the number of 2-cycles in delta. Let [n](q)=1+q+...+q(n-1) and let ((n)(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 [GRAPHICS] (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((n)(2)) graded EL-shellable poset whose order complex triangulates a ball. The rank of delta is an element ofF(2n) is given by wt(delta) and the rank generating function is [1](q)[3](q)...[2n-1](q).