Twins in words and long common subsequences in permutations

A large family of words must contain two words that are similar. We investigate several problems where the measure of similarity is the length of a common subsequence. We construct a family of n^{1/3} permutations on n letters, such that LCS of any two of them is only cn^{1/3}, improving a construction of Beame, Blais, and Huynh-Ngoc. We relate the problem of constructing many permutations with small LCS to the twin word problem of Axenovich, Person and Puzynina. In particular, we show that every word of length n over a k-letter alphabet contains two disjoint equal subsequences of length cnk^{-2/3}. Many problems are left open.Comment: 18+epsilon page

Universal arrays

A word on $q$ symbols is a sequence of letters from a fixed alphabet of size $q$. For an integer $k\ge 1$, we say that a word $w$ is $k$-universal if, given an arbitrary word of length $k$, one can obtain it by removing entries from $w$. It is easily seen that the minimum length of a $k$-universal word on $q$ symbols is exactly $qk$. We prove that almost every word of size $(1+o(1))c_qk$ is $k$-universal with high probability, where $c_q$ is an explicit constant whose value is roughly $q\log q$. Moreover, we show that the $k$-universality property for uniformly chosen words exhibits a sharp threshold. Finally, by extending techniques of Alon [Geometric and Functional Analysis 27 (2017), no. 1, 1--32], we give asymptotically tight bounds for every higher dimensional analogue of this problem.Comment: 12 page

On abelian saturated infinite words

Let f:Z+→R be an increasing function. We say that an infinite word w is abelian f(n)-saturated if each factor of length n contains Θ(f(n)) abelian nonequivalent factors. We show that binary infinite words cannot be abelian n2-saturated, but, for any ε>0, they can be abelian n2−ε-saturated. There is also a sequence of finite words (wn), with |wn|=n, such that each wn contains at least Cn2 abelian nonequivalent factors for some constant C>0. We also consider saturated words and their connection to palindromic richness in the case of equality and k-abelian equivalence.</p