Three Classes of Pattern-Avoiding Weak Ascent Sequences

A weak ascent sequence is a word (Formula presented.) over the set of nonnegative integers satisfying (Formula presented.) and (Formula presented.) for (Formula presented.), where (Formula presented.) denotes the number of weak ascents in the word (Formula presented.); that is, the number of adjacent pairs (Formula presented.) such that (Formula presented.). In this paper, we present enumerations of three classes of weak ascent sequences avoiding a pattern of length four. More precisely, we describe each class by a generating tree with two or three labels and then, using several experimental computer techniques beyond the kernel method, we obtain an explicit formula for the generating function counting weak ascent sequences of length (Formula presented.) avoiding either 0123, 0012, or 0021.