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 $Z_3 = abacaba$.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2015
- DOI:
- 10.48550/arXiv.1510.03917
- arXiv:
- arXiv:1510.03917
- Bibcode:
- 2015arXiv151003917C
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 25 pages