(k, λ)-anti-powers and other patterns in words

Given a word, we are interested in the structure of its contiguous subwords split into k blocks of equal length, especially in the homogeneous and anti-homogeneous cases. We introduce the notion of (mu(1), ... ,mu(k))-block-patterns, words of the form w = w(1), ... ,w(k) where, when {w(1), ... ,w(k)} is partitioned via equality, there are mu(s) sets of size s for each s is an element of {1, ... ,k}. This is a generalization of the well-studied k-powers and the k-anti-powers recently introduced by Fici, Restivo, Silva, and Zamboni, as well as a refinement of the (k, lambda)-anti-powers introduced by Defant. We generalize the anti-Ramsey-type results of Fici et al. to (mu(1), ... ,mu(k))-block-patterns and improve their bounds on N-alpha(k, k), the minimum length such that every word of length N-alpha(k, k) on an alphabet of size a contains a k-power or k-anti-power. We also generalize their results on infinite words avoiding k-anti-powers to the case of (k, lambda)-anti-powers. We provide a few results on the relation between a and N-alpha(k , k) and find the expected number of (mu(1), ... ,mu(k))-block-patterns in a word of length n.