Metrics on permutations with the same peak set

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

doi:10.2140/involve.2024.17.835

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Metrics on permutations with the same peak set

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

Vol. 17 (2024), No. 5, 835–844

DOI: 10.2140/involve.2024.17.835

Abstract

Let $S_{n}$ be the symmetric group on the set ${1, 2, \dots, n}$ . Given a permutation $σ = σ_{1} σ_{2} \dots σ_{n} \in S_{n}$ , we say it has a peak at index $i$ if $σ_{i - 1} < σ_{i} > σ_{i + 1}$ . Let $Peak (σ)$ be the set of all peaks of $σ$ and define $P (S; n) = {σ \in S_{n} : Peak (σ) = S}$ . We study the Hamming metric, $ℓ_{\infty}$ -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 $S_{n}$ .

Keywords

permutations, peaks, Hamming metric, Kendall tau metric, L-infinity metric

Mathematical Subject Classification

Primary: 05A05

Secondary: 05A15

Milestones

Received: 23 August 2023

Revised: 5 January 2024

Accepted: 8 January 2024

Published: 21 November 2024

Communicated by Steven J. Miller

Authors

Alexander Diaz-Lopez
Department of Mathematics and Statistics Villanova University Villanova, PA United States

Kathryn Haymaker
Department of Mathematics and Statistics Villanova University Villanova, PA United States

Kathryn Keough
Department of Mathematics and Statistics Villanova University Villanova, PA United States

Jeongbin Park
Department of Mathematics and Statistics Villanova University Villanova, PA United States

Edward White
Department of Mathematics University of Iowa Iowa City, IA United States

Vol. 17, No. 5, 2024