Multiple twins in permutations

TL;DR

This paper investigates the size of the largest r-tuplets in permutations, establishing bounds and conjecturing the precise order of magnitude, thereby generalizing prior work on the case r=2.

Contribution

The paper introduces bounds for the size of r-tuplets in permutations and conjectures the bounds are tight, extending previous results from r=2 to general r.

Findings

Established upper bound: $t^{(r)}(n)=O(n^{r/(2r-1)})$

Established lower bound: $t^{(r)}(n)= ext{Omega}(n^{R/(2R-1)})$ where $R=\binom{2r-1}{r}$

Proved bounds hold for almost all permutations

Abstract

By an $r$-tuplet in a permutation we mean a family of $r$ pairwise disjoint subsequences with the same relative order. The length of an $r$-tuplet is defined as the length of any single subsequence in the family. Let $t^{(r)}(n)$ denote the largest $k$ such that every permutation of length $n$ contains an $r$-tuplet of length $k$. We prove that $t^{(r)}(n)=O\left(n^{\frac r{2r-1}}\right)$ and $t^{(r)}(n)=\Omega\left( n^{\frac{R}{2R-1}} \right)$, where $R=\binom{2r-1}r$. We conjecture that the upper bound brings the correct order of magnitude of $t^{(r)}(n)$ and support this conjecture by proving that it holds for almost all permutations. Our work generalizes previous studies of the case $r=2$.