Variations on twins in permutations

Abstract

Let π be a permutation of the set rns " t1, 2, . . . , nu. Two disjoint orderisomorphic subsequences of π 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 Ωp ? nq. On the other hand, by a simple probabilistic argument Gawron proved that for every n ě 1 there exist permutations with all twins having length Opn 2{3 q.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 Ωpn 2{3 q. This was also proved recently by Bukh and Rudenko (see Remark 1.2 below on the interrelation between these two results).We also 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 τ , then the corresponding extremal function equals Θp ? nq, provided that τ is not monotone. In case of block twins (each twin occupies a segment) we prove that it is p1 `op1qq 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.

Isomorphisms between random graphs

“…A combinatorial application of this problem is studied in [9]. Similar questions can be asked for other combinatorial objects (largest disjoint isomorphic subtrees in a tree, largest disjoint order isomorphic pair of sub-permutations in a permutation -see [6] for applications of such problems).…”

Tight multiple twins in permutations

“…Note that the containment of tight twins, and, in particular, order repetitions, is not monotone in the sense that the absence of tight twins of length k does not exclude the presence of tight twins longer than k. By using the probabilistic method, we proved in [4], Thm. 3.6, that there exist arbitrarily long permutations without tight twins longer than 12.…”

“…Let tt prq pnq denote the largest integer k such that every permutation of length n contains tight r-twins of length k. Our result from [4], mentioned above, states that tt p2q pnq ď 12.…”

“…Large pairs of similar sub-permutations (called twins) in a given, or random, permutation have recently attracted some attention (cf. [8], [3], [4]). Here we are exclusively devoted to twins which appear in blocks.…”

“…Most likely this constant is not optimal, but it cannot go all the way down to 1, as every permutation of length 6 contains tight twins of length 2 (see [4], Prop. 3.7).…”