On alternating signed permutations with the maximal number of fixed points

TL;DR

This paper extends a known result about alternating permutations with many fixed points being equidistributed with derangements from type A to type B, involving signed permutations.

Contribution

It generalizes the equidistribution result from type A to type B for alternating signed permutations with maximal fixed points.

Findings

Proves equidistribution between certain classes of signed permutations and derangements of type B.

Extends Stanley's conjecture from type A to type B.

Provides combinatorial proofs for the generalized results.

Abstract

A conjecture by R. Stanley on a class of alternating permutations, which is proved by R. Chapman and L. Williams states that alternating permutations with the maximal number of fixed points is equidistributed with derangements. We extend this (type $A$) result to type $B$: We prove that various classes of alternating signed permutations with the maximal number of fixed points is equidistributed with certain types of derangements (of type $B$), respectively.