Some enumerative results related to ascent sequences

An ascent sequence is one consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it in the sequence. Ascent sequences have recently been shown to be related to (2+2)-free posets and a variety of other combinatorial structures. In this paper, we prove in the affirmative some recent conjectures concerning pattern avoidance for ascent sequences. Given a pattern $\tau$, let ${\mathcal{S}}_{\tau }(n)$ denote the set of ascent sequences of length $n$ avoiding $\tau$. Here, we show that the joint distribution of the statistic pair $(\text{asc},\text{zero})$ on ${\mathcal{S}}_{0012}(n)$ is the same as $(\text{asc},\text{RLm})$ on the set of 132-avoiding permutations of length $n$. In particular, the ascent statistic on ${\mathcal{S}}_{0012}(n)$ has the Narayana distribution. We also enumerate $S_{\tau }(n)$ when $\tau =1012$ and $\tau =0123$ and confirm the conjectured formulas in these cases. We combine combinatorial and algebraic techniques to prove our results, in two cases, making use of the kernel method. Finally, we discuss the case of avoiding 210 and determine two related recurrences.