Tight multiple twins in permutations

TL;DR

This paper investigates the maximum length of tight multiple twins in permutations, focusing on the guaranteed size in both arbitrary and random permutations, revealing structural properties of permutation patterns.

Contribution

It introduces the concept of tight multiple twins in permutations and analyzes their maximal length, providing new insights into permutation structure and pattern repetition.

Findings

Established bounds for the longest tight r-twins in permutations

Analyzed the expected size of tight twins in random permutations

Identified structural properties influencing the formation of tight twins

Abstract

Two permutations are similar if they have the same length and the same relative order. A collection of $r\ge2$ disjoint, similar subsequences of a permutation $\pi$ form $r$-twins in $\pi$. We study the longest guaranteed length of $r$-twins which are tight in the sense that either each twin alone forms a block or their union does. We address the same question with respect to a random permutation.