Alternating, Pattern-Avoiding Permutations - Lewis, Joel Brewster (The Electronic Journal of Combinatorics, 2009)
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.