## Joshua Cooper ; Danny Rorabaugh - Asymptotic Density of Zimin Words

dmtcs:1302 - Discrete Mathematics & Theoretical Computer Science, March 17, 2016, Vol. 18 no. 3 - https://doi.org/10.46298/dmtcs.1302
Asymptotic Density of Zimin Words

Authors: Joshua Cooper ; Danny Rorabaugh

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 $Z_3 = abacaba$.

Volume: Vol. 18 no. 3
Section: Combinatorics
Published on: March 17, 2016
Submitted on: March 17, 2016
Keywords: Mathematics - Combinatorics