  We say that a permutation $\pi$ is a Motzkin permutation if it avoids 132 and
there do not exist $a<b$ such that $\pi_a<\pi_b<\pi_{b+1}$. We study the
distribution of several statistics in Motzkin permutations, including the
length of the longest increasing and decreasing subsequences and the number of
rises and descents. We also enumerate Motzkin permutations with additional
restrictions, and study the distribution of occurrences of fairly general
patterns in this class of permutations.
