On the number of collinear triples in permutations

TL;DR

This paper provides a direct proof for a lower bound on the minimum number of collinear triples in permutations over finite fields and calculates the expected number of such triples.

Contribution

It offers a direct proof of a known bound and determines the expected number of collinear triples in permutations over finite fields.

Findings

Established a direct proof for the lower bound on collinear triples.

Calculated the expected number of collinear triples in permutations.

Confirmed the bound for prime number moduli.

Abstract

Let $\alpha:\mathbb{Z}_n\to\mathbb{Z}_n$ be a permutation and $\Psi(\alpha)$ be the number of collinear triples modulo $n$ in the graph of $\alpha$. Cooper and Solymosi had given by induction the bound $\min_{\alpha}\Psi(\alpha)\geq\lceil(n-1)/4\rceil$ when $n$ is a prime number. The main purpose of this paper is to give a direct proof of that bound. Besides, the expected number of collinear triples a permutation can have is also been determined.