Generating Trees and Pattern Avoidance in Alternating Permutations

Abstract

We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern $2143$.  We use a generating tree approach to construct a recursive bijection between the set $A_{2n}(2143)$ of alternating permutations of length $2n$ avoiding $2143$ and the set of standard Young tableaux of shape $\langle n, n, n\rangle$, and between the set $A_{2n + 1}(2143)$ of alternating permutations of length $2n + 1$ avoiding $2143$ and the set of shifted standard Young tableaux of shape $\langle n + 2, n + 1, n\rangle$.  We also give a number of conjectures and open questions on pattern avoidance in alternating permutations and generalizations thereof.