Variations on Twins in Permutations

Abstract

Let $\pi$ be a permutation of the set $[n]=\{1,2,\dots, n\}$. Two disjoint order-isomorphic subsequences of $\pi$ are called twins. How long twins are contained in every permutation? The well known Erdős-Szekeres theorem implies that there is always a pair of twins of length $\Omega(\sqrt{n})$. On the other hand, by a simple probabilistic argument Gawron proved that for every $n\geqslant 1$ there exist permutations with all twins having length $O(n^{2/3})$. He conjectured that the latter bound is the correct size of the longest twins guaranteed in every permutation. We support this conjecture by showing that almost all permutations contain twins of length $\Omega(n^{2/3}/\log n^{1/3})$. Recently, Bukh and Rudenko have tweaked our proof and removed the log-factor. For completeness, we also present our version of their proof (see Remark 2 below on the interrelation between the two proofs).
 In addition, we study several variants of the problem with diverse restrictions imposed on the twins. For instance, if we restrict attention to twins avoiding a fixed permutation $\tau$, then the corresponding extremal function equals $\Theta(\sqrt{n})$, provided that $\tau$ is not monotone. In case of block twins (each twin occupies a segment) we prove that it is $(1+o(1))\frac{\log n}{\log\log n}$, while for random permutations it is twice as large. For twins that jointly occupy a segment (tight twins), we prove that for every $n$ there are permutations avoiding them on all segments of length greater than $24$.

Decomposing random permutations into order-isomorphic subpermutations

“…The proof of Gawron's upper bound comes from applying the first moment method to a uniformly random permutation, and there has since been work on finding a matching lower bound for random permutations. A lower bound of Ω n 2/3 / log 1/3 (n) was shown by Dudek, Grytczuk and Ruciński [13], and this was improved to the sharp bound of Ω(n 2/3 ) by Bukh and Rudenko [4]. There has also been further work of Dudek, Grytczuk and Ruciński in the same vein concerning k-twins [11], which are a collection of k pairwise disjoint orderisomorphic subpermutations of a single permutation, and other notions of twins with additional restrictions imposed or a weaker similarity condition (see [9,12]).…”

“…In the following, the idea of constructing an auxiliary bipartite multigraph has previously been used by Dudek, Grytczuk and Ruciński [13] and later by Bukh and Rudenko [4] to study twins in a single random permutation. The use of a Poisson process to sample random permutations also appears in [4].…”

“…The analogous problem for permutations was raised by Dudek, Grytczuk and Ruciński [13]. Given a sequence a = a 1 , .…”

“…Following Dudek, Grytczuk and Ruciński [13], we say that two permutations σ and π are ℓ-similar if they can each be decomposed into ℓ subpermutations σ (1) , . .…”

“…However, it is far from obvious what happens on average. Problem 1.1 (Dudek, Grytczuk and Ruciński [13]). Let σ and π be two permutations of length n chosen independently and uniformly at random.…”