Prolific permutations and permuted packings: downsets containing many large patterns

TL;DR

This paper characterizes k-prolific permutations, showing their existence for all sufficiently large sizes and establishing a bijection with permuted diamond packings, advancing understanding of permutation pattern uniqueness.

Contribution

It provides a complete characterization of k-prolific permutations and introduces a bijection with permuted packings, revealing new structural insights.

Findings

k-prolific permutations exist for all sizes m q k^2/2+2k+1

No k-prolific permutations exist below this size threshold

A bijection links k-prolific permutations to permuted diamond packings

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 \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.