Geodesic
Asymptotic Density of Zimin Words
Discrete mathematics & theoretical computer science, Tome 18 (2015-2016) no. 3.

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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 $Z_3 = abacaba$. 
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     author = {Cooper, Joshua and Rorabaugh, Danny},
     title = {Asymptotic {Density} of {Zimin} {Words}},
     journal = {Discrete mathematics & theoretical computer science},
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Cooper, Joshua; Rorabaugh, Danny. Asymptotic Density of Zimin Words. Discrete mathematics & theoretical computer science, Tome 18 (2015-2016) no. 3. doi : 10.46298/dmtcs.1302. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.1302/

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