How Balanced Can Permutations Be?

TL;DR

This paper investigates the existence and construction of permutations that are perfectly balanced with respect to smaller permutation patterns, providing explicit constructions for small cases and proving non-existence for larger ones.

Contribution

It explicitly constructs $k$-balanced permutations for $k \\le 3$ and all suitable $n$, and proves non-existence for $k \\ge 4$, establishing bounds on how far permutations are from being balanced.

Findings

Explicit constructions for $k$-balanced permutations when $k \\le 3$.

Proof of non-existence of such permutations for $k \\ge 4$.

Lower bounds on the distance from being $k$-balanced for larger $k$.

Abstract

A permutation $\pi \in \mathbb{S}_n$ is $k$-balanced if every permutation of order $k$ occurs in $\pi$ equally often, through order-isomorphism. In this paper, we explicitly construct $k$-balanced permutations for $k \le 3$, and every $n$ that satisfies the necessary divisibility conditions. In contrast, we prove that for $k \ge 4$, no such permutations exist. In fact, we show that in the case $k \ge 4$, every $n$-element permutation is at least $\Omega_n(n^{k-1})$ far from being $k$-balanced. This lower bound is matched for $k=4$, by a construction based on the Erd\H{o}s-Szekeres permutation.