The k -regular sequences form a large class studied in number theory, combinatorics, and other parts of discrete mathematics. This class is known to be closed under many natural operations, such as term-by-term sum, product, running sum, and so forth, but it is not closed under running maximum. Proving the previously-known counterexample, involving the Stern sequence, required intricate arguments. In this note, we construct a significantly simpler example of a k -regular sequence whose running maximum is not k -regular.

A length n word is (palindromic) rich if it contains the maximum possible number, which is n , of distinct non-empty palindromic factors. We prove both necessary and sufficient conditions for richness in terms of run-length encodings of words. Relating sufficient conditions to integer partitions, we prove a lower bound of order C n , where C ≈ 37.6 , on the growth function of the language of binary rich words. From experimental study we suggest that this growth function actually grows more slowly than n n , which makes our lower bound quite reasonable.