Prolific permutations and permuted packings: Downsets containing many large patterns

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 kprolific 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.

On the poset of non-attacking king permutations

“…By removing from π the digit 8 (or the digit 7) and standardizing, we get ∇ * 8 (π) = τ = [3, 5, 1, 4, 6, 2, 7] ∈ K 7 . This implies the existence of a regent of σ (for example [5,1,3,6,2,8,4,7] The permutations π ∈ K n which have no regents have a special structure. An example for such an element is π = [7, 5, 8, 6, 2, 4, 1, 3, 10, 12, 9, 11] ∈ K 12 .…”

“…Moreover, by removing any value from π we get a 3−block. (by way of illustration, if we remove the digit 8 and standardise we get [7,5,6,2,4,1,3,9,11,8,10]).…”

“…We are interested in the sub-poset K = ∪ n∈N K n containing only the king permutations. Its minimal element is the identity permutation [1], and in the next level appear [2413] and [3142].…”

The minimum Manhattan distance and minimum jump of permutations

“…Note that for n ≥ 2 we have d(π) ≥ 2 for all π ∈ S n . The minimum Manhattan distance is a natural measure when thinking of a permutation as its graph, but was first studied (under the name of the breadth of a permutation) by Bevan, Homberger, and Tenner [2] in the context of permutation patterns. We now briefly explain this context.…”

“…For a given sequence a of length n, we define the standardization of a to be the unique sequence on the letters [n] which is in the same relative order as a. The pattern ordering imposes a partial order on the set of all permutations: For π ∈ S n and σ ∈ S k , we say that π contains σ as a pattern (denoted σ ≺ π) if there is a subsequence of π(1)π (2) . .…”