Metrics on Signed Permutations with the Same Peak Set

TL;DR

This paper extends the study of permutation metrics to signed permutations of type B, analyzing Hamming, infinity, and word metrics on subsets with a fixed peak set, and determining their extremal values.

Contribution

It generalizes previous work by considering signed permutations and multiple metrics, providing new insights into their extremal metric values within fixed peak sets.

Findings

Derived formulas for minimum and maximum metric values

Extended metrics analysis to signed permutations of type B

Identified extremal values for Hamming, infinity, and word metrics

Abstract

Let $S^B_n$ be the Coxeter group of type B. We denote the set of indices where $\sigma\in S^B_n$ has a peak as $Peak(\sigma)$ and let $P^{B}(S;n)=\{\sigma \in S^{B}_n~|~ Peak(\sigma)=S\}$. In \cite{metrics}, Diaz-Lopez, Haymaker, Keough, Park and White considered metrics for unsigned permutations with the same peak set. In this paper, we generalize their result by studying Hamming, $l_{\infty}$, and the word metrics on $P^{B}(S;n)$ for all $S$. We also determine the minimum and maximum possible values that these metrics can achieve in these subsets of $S^B_n$.