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Density maximizers of layered permutations

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KABELA Adam KRÁĽ Daniel NOEL Jon Andrew PIERRON Théo

Rok publikování 2022
Druh Článek v odborném periodiku
Časopis / Zdroj Electronic Journal of Combinatorics
Fakulta / Pracoviště MU

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Citace
www https://www.combinatorics.org/ojs/index.php/eljc/article/view/v29i3p56
Doi http://dx.doi.org/10.37236/10781
Klíčová slova permutations; layered permutations
Popis A permutation is layered if it contains neither 231 nor 312 as a pattern. It is known that, if ? is a layered permutation, then the density of ? in a permutation of order n is maximized by a layered permutation. Albert, Atkinson, Handley, Holton and Stromquist [Electron. J. Combin. 9 (2002), #R5] claimed that the density of a layered permutation with layers of sizes (a, 1, b) where a, b > 2 is asymptotically maximized by layered permutations with a bounded number of layers, and conjectured that the same holds if a layered permutation has no consecutive layers of size one and its first and last layers are of size at least two. We show that, if ? is a layered permutation whose first layer is sufficiently large and second layer is of size one, then the number of layers tends to infinity in every sequence of layered permutations asymptotically maximizing the density of ?. This disproves the conjecture and the claim of Albert et al. We complement this result by giving sufficient conditions on a layered permutation to have asymptotic or exact maximizers with a bounded number of layers.
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