An Upper Bound on the Number of $(132,213)$-Avoiding Cyclic Permutations

TL;DR

This paper establishes the first nontrivial upper bound on the number of cyclic permutations avoiding the patterns 132 and 213, and provides an algorithm to identify such permutations based on layer lengths.

Contribution

It introduces a new upper bound of n^2 * 2^{n/2} for these permutations and an algorithm leveraging layer lengths for cyclicity testing.

Findings

Upper bound of n^2 * 2^{n/2} on the count of (132,213)-avoiding cyclic permutations

First nontrivial upper bound for this permutation class

Algorithm for cyclicity detection based on layer lengths

Abstract

We show a $n^2 \cdot 2^{n/2}$ upper bound on the number of $(132,213)$ avoiding cyclic permutations. This is the first nontrivial upper bound on the number of such permutations. We also construct an algorithm to determine whether a $(132,213)$ avoiding permutation is cyclic that references only the permutation's layer lengths.