How Balanced Can Permutations Be?

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A permutation π∈Sn is k-balanced if every permutation of order k occurs in π equally often, through order-isomorphism. In this paper, we explicitly construct k-balanced permutations for k≤3, and every n that satisfies the necessary divisibility conditions. In contrast, we prove that for k≥4, no such permutations exist. In fact, we show that in the case k≥4, every n-element permutation is at least Ωn(nk-1) far from being k-balanced. This lower bound is matched for k=4, by a construction based on the Erdős–Szekeres permutation.