Prolific permutations and permuted packings: downsets containing many   large patterns
  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 \ge 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. 
David Bevan, Cheyne Homberger, and Bridget Eileen Tenner
math.CO