Pinnacle sets of signed permutations

TL;DR

This paper extends the concept of pinnacle sets from permutations to signed permutations of types B and D, providing formulas and enumerative results for admissible pinnacle sets in these groups.

Contribution

It introduces the definition of pinnacle sets for signed permutations of types B and D and derives closed formulas for counting admissible pinnacle sets.

Findings

Closed formula for the number of admissible pinnacle sets in types B and D

Enumeration of pinnacle sets for signed permutations

Extension of pinnacle set theory to signed permutation groups

Abstract

Pinnacle sets record the values of the local maxima for a given family of permutations. They were introduced by Davis-Nelson-Petersen-Tenner as a dual concept to that of peaks, previously defined by Billey-Burdzy-Sagan. In recent years pinnacles and admissible pinnacles sets for the type $A$ symmetric group have been widely studied. In this article we define the pinnacle set of signed permutations of types $B$ and $D$. We give a closed formula for the number of type $B$/$D$ admissible pinnacle sets and answer several other related enumerative questions.