Asymptotic Density of Zimin Words.

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 I-n (V, [q]) be the probability that a random length n word on the alphabet [q] = {1, 2, ... q} is an instance of V. Having previously shown that lim(n ->infinity) I-n (V, [q]) exists, we now calculate this limit for two Zimin words, Z(2) = aba and Z(3) = abacaba.