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.

如果阶数 k 的每个排列通过阶同构在 \（\pi \） 中出现的频率相同，则排列 \（\pi \in \mathbb {S}_n\） 是 k 平衡的。在本文中，我们为 \（k \le 3\） 以及满足必要整除条件的每个 n 显式构造了 k 平衡排列。相反，我们证明对于 \（k \ge 4\），不存在这样的排列。事实上，我们表明，在 \（k \ge 4\） 的情况下，每个 n 元素排列至少与 \（\Omega _n（n^{k-1}）\） 相去甚远。这个下限与\（k=4\）匹配，由基于Erdős-Szekeres排列的构造匹配。