  We study the problem of counting alternating permutations avoiding
collections of permutation patterns including 132. We construct a bijection
between the set S_n(132) of 132-avoiding permutations and the set A_{2n +
1}(132) of alternating, 132-avoiding permutations. For every set p_1, ..., p_k
of patterns and certain related patterns q_1, ..., q_k, our bijection restricts
to a bijection between S_n(132, p_1, ..., p_k), the set of permutations
avoiding 132 and the p_i, and A_{2n + 1}(132, q_1, ..., q_k), the set of
alternating permutations avoiding 132 and the q_i. This reduces the enumeration
of the latter set to that of the former.
