  In this paper, we characterize and enumerate pattern-avoiding permutations
composed of only 3-cycles. In particular, we answer the question for the six
patterns of length 3. We find that the number of permutations composed of $n$
3-cycles that avoid the pattern 231 (equivalently 312) is given by $3^{n-1}$,
while the generating function for the number of those that avoid the pattern
132 (equivalently 213) is given by a formula involving the generating functions
for the well-known Motzkin numbers and Catalan numbers. The number of
permutations composed of $n$ 3-cycles that avoid the pattern 321 is
characterized by a weighted sum involving statistics on Dyck paths of
semilength~$n$.
