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Metrics on permutations with the same peak set

Alexander Diaz-Lopez, Kathryn Haymaker, Kathryn Keough, Jeongbin Park, Edward White

Abstract

Let $S_n$ be the symmetric group on the set $\{1,2,\ldots,n\}$. Given a permutation $σ=σ_1σ_2 \cdots σ_n \in S_n$, we say it has a peak at index $i$ if $σ_{i-1}<σ_i>σ_{i+1}$. Let $\text{Peak}(σ)$ be the set of all peaks of $σ$ and define $P(S;n)=\{σ\in S_n\, | \,\text{Peak}(σ)=S\}$. In this paper we study the Hamming metric, $\ell_\infty$-metric, and Kendall-Tau metric on the sets $P(S;n)$ for all possible $S$, and determine the minimum and maximum possible values that these metrics can attain in these subsets of $S_n$.

Metrics on permutations with the same peak set

Abstract

Let be the symmetric group on the set . Given a permutation , we say it has a peak at index if . Let be the set of all peaks of and define . In this paper we study the Hamming metric, -metric, and Kendall-Tau metric on the sets for all possible , and determine the minimum and maximum possible values that these metrics can attain in these subsets of .
Paper Structure (4 sections, 9 theorems, 17 equations, 3 tables)

This paper contains 4 sections, 9 theorems, 17 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Metrics on $S_n$
  3. Maximum and minimum distances among permutations with the same peak set
  4. Acknowledgements

Key Result

Lemma 2.3

The Kendall-Tau metric is right invariant, that is, for any $\sigma,\tau, \alpha\in S_n,$$d_K(\sigma, \rho)=d_K(\sigma\alpha, \rho\alpha)$.

Theorems & Definitions (22)

  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Definition 2.7
  • ...and 12 more