Alternating, pattern-avoiding permutations

TL;DR

This paper establishes a bijection between 132-avoiding permutations and alternating, 132-avoiding permutations, simplifying their enumeration by relating these two classes through pattern avoidance constraints.

Contribution

It introduces a bijection that connects 132-avoiding permutations with alternating, 132-avoiding permutations, extending to sets avoiding additional patterns, thus aiding enumeration.

Findings

Bijection between S_n(132) and A_{2n+1}(132)

Reduction of enumeration problems for pattern-avoiding alternating permutations

Framework for extending pattern avoidance bijections to multiple patterns

Abstract

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.