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Prolific permutations and permuted packings : downsets containing many large patterns

Bevan, David and Homberger, Cheyne and Tenner, Bridget Eileen (2017) Prolific permutations and permuted packings : downsets containing many large patterns. Journal of Combinatorial Theory Series A. ISSN 0097-3165 (In Press)

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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.