# Prolific permutations and permuted packings: Downsets containing many large patterns

with
David Bevan
Bridget Tenner

## Abstract:

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.