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@article{2245922,
author = {Kabela, Adam and Kráľ, Daniel and Noel, Jon Andrew and Pierron, Théo},
article_number = {3},
doi = {http://dx.doi.org/10.37236/10781},
keywords = {permutations; layered permutations},
language = {eng},
issn = {1077-8926},
journal = {Electronic Journal of Combinatorics},
title = {Density maximizers of layered permutations},
url = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/v29i3p56},
volume = {29},
year = {2022}
}
TY - JOUR
ID - 2245922
AU - Kabela, Adam - Kráľ, Daniel - Noel, Jon Andrew - Pierron, Théo
PY - 2022
TI - Density maximizers of layered permutations
JF - Electronic Journal of Combinatorics
VL - 29
IS - 3
SP - 1-21
EP - 1-21
SN - 10778926
KW - permutations
KW - layered permutations
UR - https://www.combinatorics.org/ojs/index.php/eljc/article/view/v29i3p56
N2 - A permutation is layered if it contains neither 231 nor 312 as a pattern. It is known that, if σ is a layered permutation, then the density of σ 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 σ 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 σ. 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.
ER -
KABELA, Adam, Daniel KRÁĽ, Jon Andrew NOEL and Théo PIERRON. Density maximizers of layered permutations. \textit{Electronic Journal of Combinatorics}. 2022, vol.~29, No~3, p.~1-21. ISSN~1077-8926. Available from: https://dx.doi.org/10.37236/10781.
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