Results on long twins in random words and permutations
Elliott Liu, Linus Tang, Jessica Wan
TL;DR
The paper investigates long twin-like structures in random words and random permutations. It derives sharp one-sided tail bounds showing that the maximum $r$-power length in a random word over $[k]^n$ concentrates around $ rac{ ext{log} n}{(r-1) ext{log} k}$ and establishes concentration of its expectation. In random permutations, it proves that for fixed $k$ and $r o ty$, a uniform permutation of $[rk]$ almost surely contains $r$ disjoint increasing subsequences of length $k$, generalizing prior $k=2$ results, and it improves the lower bound on the length of alternating twins to $igl( frac{1}{3}+0.0989- ext{o}(1)igr)n$ via computer-assisted analysis. A further generalization shows that the number of tight $r$-twins in random permutations of length $rk$ fills the permutation as $r o ty$, and the paper collects the appendix with pseudocode for the alternating-twins computation. Overall, the work sharpens probabilistic understanding of twin-like structures in random discrete objects and provides precise asymptotics and constants for core twin phenomena.
Abstract
We study long $r$-twins in random words and permutations. Motivated by questions posed in works of Dudek-Grytczuk-Ruciński, we obtain the following. For a uniform word in $[k]^n$ we prove sharp one-sided tail bounds showing that the maximum $r$-power length (the longest contiguous block that can be partitioned into $r$ identical subblocks) is concentrated around $\frac{\log n}{(r-1)\log k}$. For random permutations, we prove that for fixed $k$ and $r\to\infty$, a uniform permutation of $[rk]$ a.a.s. contains $r$ disjoint increasing subsequences of length $k$, generalizing a previous result that proves this for $k=2$. Finally, we use a computer-aided pattern count to improve the best known lower bound on the length of alternating twins in a random permutation to $α_n \ge \left(\tfrac{1}{3}+0.0989-o(1)\right)n$, strengthening the previous constant.