Wilf classes for weak ascent sequences avoiding a pair or triple of length-3 patterns

Abstract

A weak ascent sequence is a word $\pi ={\pi }_1{\pi }_2\cdots {\pi }_n$ over the set of nonnegative integers such that ${\pi }_1=0$ and ${\pi }_i\le 1+\text{weak}\_\text{asc}({\pi }_1{\pi }_2\cdots {\pi }_{i-1})$ for $i=2,\dots ,n$, where $\text{weak}\_\text{asc}({\pi }_1{\pi }_2\cdots {\pi }_m)$ is the number of weak ascents in the word ${\pi }_1{\pi }_2\cdots {\pi }_m$, that is, the number of two-entry factors ${\pi }_j{\pi }_{j+1}$ such that ${\pi }_j\le {\pi }_{j+1}$. Here we obtain some enumerative results for weak ascent sequences avoiding a set of two or three 3-letter patterns, leading to a conjecture for the number of Wilf equivalence classes for weak ascent sequences avoiding a pair (respectively, triple) of 3-letter patterns. The main tool is the use of generating trees. Some cases are treated using bijective methods.