Cheyne Homberger

Prolific permutations and permuted packings: Downsets containing many large patterns

with David Bevan Bridget Tenner


A permutation of n letters is k-prolific if each (n-k)-subset of the letters in its one-line notation forms a unique pattern. We present a complete characterization of k-prolific permutations for each k, proving that $k$-prolific permutations of m letters exist for every m <= k^2/2+2k+1, and that none exist of smaller size. Key to these results is a natural bijection between k-prolific permutations and certain ``permuted'' packings of diamonds.