  We present a method, illustrated by several examples, to find explicit counts
of permutations containing a given multiset of three letter patterns. The
method is recursive, depending on bijections to reduce to the case of a smaller
multiset, and involves a consideration of separate cases according to how the
patterns overlap. Specifically, we use the method (i) to provide combinatorial
proofs of Bona's formula {2n-3}choose{n-3} for the number of n-permutations
containing one 132 pattern and Noonan's formula 3/n {2n}choose{n+3} for one 123
pattern, (ii) to express the number of n-permutations containing exactly k 123
patterns in terms of ballot numbers for k<=4, and (iii) to express the number
of 123-avoiding n-permutations containing exactly k 132 patterns as a linear
combination of powers of 2, also for k<=4. The results strengthen the
conjecture that the counts are algebraic for all k.
