A Generalisation of a Result on Monotone Arithmetic Progressions in Permutations of the Positive Integers

TL;DR

This paper generalizes a previous result by constructing permutations of positive integers that avoid certain monotone arithmetic progressions with specific common differences, extending the understanding of pattern avoidance in permutations.

Contribution

It introduces a broad generalization showing the existence of permutations avoiding monotone arithmetic progressions with common differences not divisible by powers of two.

Findings

Existence of permutations avoiding progressions with odd differences

Extension of previous specific case to general powers of two

Broader understanding of pattern avoidance in permutations

Abstract

A permutation of the positive integers avoiding 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 $2^k$.