A Note on 3-free Permutations

TL;DR

This paper develops a dynamic programming method to count 3-free permutations, extending known enumerations from n=20 to n=90, and refines bounds on their quantity.

Contribution

It introduces a new algorithm for counting 3-free permutations and provides extended and corrected enumeration data up to n=90.

Findings

Extended enumeration of 3-free permutations up to n=90

Corrected previous results in the literature

Improved bounds on the number of 3-free permutations

Abstract

Let $\theta(n)$ denote the number of permutations of $\{1,2,\ldots,n\}$ that do not contain a 3-term arithmetic progression as a subsequence. Such permutations are known as 3-free permutations. We present a dynamic programming algorithm to count all 3-free permutations of $\{1,2,\ldots,n\}$. We use the output to extend and correct enumerative results in the literature for $\theta(n)$ from $n=20$ out to $n=90$ and use the new values to inductively improve existing bounds on $\theta(n)$.