Excedance Numbers for the Permutations of Type B
Keywords:
type $B$ permutation, type $B$ excedance, type $B$ derangement, fixed point, cycle, Euler number, Springer number
Abstract
This work provides a study on the multidistribution of type $B$ excedances, fixed points and cycles on the permutations of type $B$. We derive the recurrences and closed formulas for the distribution of signed excedances on type $B$ permutations as well as derangements via combinatorial construction. Based on this result, we obtain the recurrence and generating function for the signed excedance polynomial and disclose some relationships with Euler numbers and Springer numbers, respectively.