A Statistic on Involutions

• Authors: Rajendra S. Deodhar;Murali K. Srinivasan
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2001

We define a statistic, called iweight, on involutions and consider two applications in which this statistic arises. Let iI(in) denote the set of all involutions on [n](=\{1,2,\ldots, n\}) and let iF(2in) denote the set of all fixed point free involutions on [2in]. For an involution δ, let |δ| denote the number of 2-cycles in δ. Let [n]_q=1+q+\cdots +q^{n-1} and let (\begin{array}{@{}c@{}}\\[-22pt] \scriptstyle n\\[-5pt] \scriptstyle k\\[-12pt]\end{array})_q denote the iq-binomial coefficient. There is a statistic wt on iI(in) such that the following results are true.(i) We have the expansion\[\Big(\begin{array}{@{}c@{}} n\\[3pt] k\end{array}\Big)_q =\sum_{\delta \in I(n)} (q-1)^{|\delta|}q^{\rm wt(\delta)} \bigg(\begin{array}{@{}c@{}} n-2|\delta|\\[3pt] k-|\delta|\end{array}\bigg). \](ii) An analog of the (strong) Bruhat order on permutations is defined on iF(2in) and it is shown that this gives a rank-2(\begin{array}{@{}c@{}}\\[-22pt] \scriptstyle n\\[-3pt] \scriptsize 2\\[-12pt]\end{array}) graded EL-shellable poset whose order complex triangulates a ball. The rank of \delta\,{\in}\,F(2n) is given by wt(δ) and the rank generating function is [1]_q[3]_q\cdots[2n-1]_q.