  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 $(\asc,\zero)$ on
$\mathcal{S}_{0012}(n)$ is the same as $(\asc,\RLm)$ 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.
