Asymptotic Density of Zimin Words

Abstract

Word $W$ is an instance of word $V$ provided there is a homomorphism $\phi$ mapping letters to nonempty words so that $\phi(V) = W$. For example, taking $\phi$ such that $\phi(c)=fr$, $\phi(o)=e$ and $\phi(l)=zer$, we see that "freezer" is an instance of "cool". Let $\mathbb{I}_n(V,[q])$ be the probability that a random length $n$ word on the alphabet $[q] = \{1,2,\cdots q\}$ is an instance of $V$. Having previously shown that $\lim_{n \rightarrow \infty} \mathbb{I}_n(V,[q])$ exists, we now calculate this limit for two Zimin words, $Z_2 = aba$ and $Z_3 = abacaba$. Comment: 25 pages

On Long Words Avoiding Zimin Patterns

“…Our lower bound on g is not as good as the one obtained by combining Theorem 9 with Corollary 2 but its proof seems more direct (already more direct than the proof of Theorem 9 itself). The proof follows a similar strategy as the (slightly better) doubly-exponential lower bound for f from [5], but again, seems to be more direct. Our novel contribution is to provide a doubly-exponential upper bound on g in Section 5.2.…”

“…Computing the exact value of f (n, k) for n ≥ 1 and k ≥ 2, or at least giving upper and lower bounds on its value, has been the topic of several articles in recent years [5,21,15,6].…”

“…For small values of n and k, known results from [15,14] In general, Cooper and Rorabaugh [5,Theorem 1.1] showed that the value of f (n, k) is upper-bounded by a tower of exponentials of height n − 1. To make this more precise let us define the tower function Tower : N × N → N inductively as follows: Tower(0, k) = 1 and Tower(n + 1, k) = k Tower(n,k) for all n, k ∈ N.…”

“…Theorem 1 (Cooper/Rorabaugh [5]) For all n ≥ 1 and k ≥ 2, f (n, k) ≤ Tower(n − 1, K), where K = 2k + 1.…”

“…To our knowledge, this is the best known lower bound for f . Theorem 2 (Cooper/Rorabaugh [5]) f (n, k) ≥ k 2 n−1 (1+o (1)) . This lower bound is obtained by estimating the expected number of occurrences of Z n in long words over a k-letter alphabet using the first moment method.…”