Counting signed derangements with right-to-left minima and excedances

Yanni Pei, Jiang Zeng

Published 2022-06-22Version 1

Alexandersson and Getachew recently proved that the multivariate enumerative polynomial of the signed derangements by excedances and right-to-left minima can be written as a sum of monomials with constant sign. We prove that each of the latter monomials is equal to the same enumerative polynomial restricted on the derangements with a fixed last element and obtain a simpler proof of their identity. We also apply a continued fraction for the enumerative polynomials of derangements with respect to numbers of right-to-left minima, excedances and cycles to confirm several observations of Alexandersson and Getachew and determine the orthogonal polynomials whose moments are the enumerative polynomials of derangements with respect to the number of right-to-left minima. We also discuss analogues formulas for the permutations of type B.