Avoiding Monotone Arithmetic Progressions in Permutations of Integers

TL;DR

This paper advances the study of permutations avoiding monotone arithmetic progressions by constructing longer progression-free permutations, improving density bounds, and generalizing previous results for specific progression lengths and differences.

Contribution

It constructs permutations avoiding longer monotone arithmetic progressions, improves density bounds, and generalizes previous results on differences and modular conditions.

Findings

Constructed permutations avoiding length 5 progressions of integers.

Improved upper and lower density results for certain permutations.

Generalized avoidance results for progressions with specific differences.

Abstract

A permutation of the integers avoiding monotone arithmetic progressions of length $6$ was constructed in (Geneson, 2018). We improve on this by constructing a permutation of the integers avoiding monotone arithmetic progressions of length $5$. We also construct permutations of the integers and the positive integers that improve on previous upper and lower density results. In (Davis et al. 1977) they constructed a doubly infinite permutation of the positive integers that avoids monotone arithmetic progressions of length $4$. We construct a doubly infinite permutation of the integers avoiding monotone arithmetic progressions of length $5$. A permutation of the positive integers that avoided monotone arithmetic progressions of length $4$ with odd common difference was constructed in (LeSaulnier and Vijay, 2011). We generalise this result and show that for each $k\geq 1$, there exists a permutation of the positive integers that avoids monotone arithmetic progressions of length $4$ with common difference not divisible by $2^k$. In addition, we specify the structure of permutations of $[1,n]$ that avoid length $3$ monotone arithmetic progressions mod $n$ as defined in (Davis et al. 1977) and provide an explicit construction for a multiplicative result on permutations that avoid length $k$ monotone arithmetic progressions mod $n$.