Alternating, Pattern-Avoiding Permutations

Abstract

We study the problem of counting alternating permutations avoiding collections of permutation patterns including 132. We construct a bijection between the set S[subscript n](132) of 132-avoiding permutations and the set A[subscript 2n+1](132) of alternating, 132-avoiding permutations. For every set p[subscript 1],…,p[subscript k] of patterns and certain related patterns q[subscript 1],…,q[subscript k], our bijection restricts to a bijection between S[subscript n](132,p[subscript 1],…,p[subscript k]), the set of permutations avoiding 132 and the p[subscript i], and A[subscript 2n+1](132,q[subscript 1],…,q[subscript k]), the set of alternating permutations avoiding 132 and the q[subscript i]. This reduces the enumeration of the latter set to that of the former.