Asymptotic Density of Zimin Words

TL;DR

This paper calculates the asymptotic probability that large random words over a finite alphabet are instances of Zimin words, specifically for Zimin words $Z_2$ and $Z_3$, extending previous existence results.

Contribution

It explicitly computes the limiting probabilities for random words being instances of Zimin words $Z_2$ and $Z_3$, advancing understanding of pattern occurrences in combinatorics on words.

Findings

Limit for $Z_2$ is approximately 0.3333.

Limit for $Z_3$ is approximately 0.0526.

Provides explicit formulas for these asymptotic probabilities.

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 $Z_3 = abacaba$.