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

TL;DR

This paper refines and relates multivariate formulas for counting signed derangements with excedances and right-to-left minima, clarifying their combinatorial meaning and extending to type B permutations.

Contribution

It provides new refinements of multivariate identities for signed derangements and connects them to continued fractions, enhancing understanding of permutation statistics.

Findings

Refined multivariate formulas for signed derangements

Clarified combinatorial interpretation of terms

Extended formulas to permutations of type B

Abstract

Recently Alexandersson and Getachew proved some multivariate generalizations of a formula for enumerating signed excedances in derangements. In this paper we first relate their work to a recent continued fraction for permutations and confirm some of their observations. Our second main result is two refinements of their multivariate identities, which clearly explain the meaning of each term in their main formulas. We also explore some similar formulas for permutations of type B.