How Balanced Can Permutations Be?

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 Erdős–Szekeres permutation.