Generating trees and pattern avoidance in alternating permutations

TL;DR

This paper extends previous work by enumerating alternating permutations avoiding the pattern 2143 using generating trees, establishing bijections with certain Young tableaux, and proposing related conjectures.

Contribution

It introduces a recursive bijection between pattern-avoiding alternating permutations and Young tableaux, expanding combinatorial enumeration methods.

Findings

Bijection between A_{2n}(2143) and standard Young tableaux of shape (n,n,n)

Bijection between A_{2n+1}(2143) and shifted standard Young tableaux of shape (n+2,n+1,n)

Formulation of conjectures and open questions on 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 standard Young tableaux of shape (n, n, n) and between the set A_{2n + 1}(2143) of alternating permutations of length 2n + 1 avoiding 2143 and shifted standard Young tableaux of shape (n + 2, n + 1, n). We also give a number of conjectures and open questions on pattern avoidance in alternating permutations and generalizations thereof.