# Prolific Permutations: Optimizing Downsets in the Pattern Poset

## 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. In this paper, we study which
values of n can produce such an object. We show that a k-prolific permutation
requires at least k^2/2+2k letters, and we construct k-prolific permutations
using only slightly more than k^2/2+2k letters, thus giving a bound for the
minimal number of letters needed to form a k-prolific permutation.