Long twins in random words

Abstract

Twins in a finite word are formed by a pair of identical subwords placed at disjoint sets of positions. We investigate the maximum length of twins in a random word over a k-letter alphabet. The obtained lower bounds for small values of k significantly improve the best estimates known in the deterministic case.Bukh and Zhou in 2016 showed that every ternary word of length n contains twins of length at least 0.34n. Our main result, with a computer-assisted proof, states that in a random ternary word of length n, with high probability, one can find twins of length at least 0.41n. In the general case, for alphabets of size k ě 3, we get analogous lower bounds of the form 1.64 k`1 n, which are better than the known deterministic bounds for k ď 354. In addition, we present similar results for multiple twins in random words.

Decomposing random permutations into order-isomorphic subpermutations

“…The answer in this case is 3n/4 + o k (n), and in particular the leading term is independent of k. Surprisingly, it turns out that this bound holds (with o k (n) replaced by O(1)) even if we wish to simultaneously decompose every graph on n vertices with the same number of edges [7]. We refer the reader to [16] for an interesting history of this problem, and mention that there are numerous variants which are actively studied (see, for instance, [2,5,10]).…”