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An ascent sequence is one consisting of non-negative integers in which thesize of each letter is restricted by the number of ascents preceding it in thesequence. Ascent sequences have recently been shown to be related to (2+2)-freeposets and a variety of other combinatorial structures. In this paper, we provein the affirmative some recent conjectures concerning pattern avoidance forascent sequences. Given a pattern $\tau$, let $\mathcal{S}_\tau(n)$ denote theset of ascent sequences of length $n$ avoiding $\tau$. Here, we show that thejoint distribution of the statistic pair $(\asc,\zero)$ on$\mathcal{S}_{0012}(n)$ is the same as $(\asc,\RLm)$ on the set of 132-avoidingpermutations 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 conjecturedformulas in these cases. We combine combinatorial and algebraic techniques toprove our results, in two cases, making use of the kernel method. Finally, wediscuss the case of avoiding 210 and determine two related recurrences.

Some enumerative results related to ascent sequences