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The s-weak order and s-permutahedra

Abstract : We introduce the s-weak order on decreasing trees, a lattice which generalizes the classical weak order on permutations. Restricting this lattice to certain trees gives rise to the s-Tamari lattice, a sublattice which generalizes the Tamari lattice. The s-weak order and the s-Tamari lattice have beautiful underlying geometric structures which we call the s-permutahedron and the s-associahedron. We provide geometric realizations of these objects in dimensions two and three, and conjecture that similar constructions exist in general. We also show that our construction is related to the ν-Tamari lattices of Préville-Ratelle and Viennot. Résumé. Nous définissons le s-ordre faible sur les arbres décroissants : un treillis qui généralise l'ordre faible sur les permutations. En restreignant à certains arbres, nous obtenons le treillis de s-Tamari, un sous-treillis qui généralise le treillis de Tamari. Ces deux objets ont de jolies structures géomé-triques sous-jacentes que nous nommons le s-permutohèdre et s-associaèdre. Nous donnons les réalisations géométriques de ces objets en dimensions deux et trois et conjecturons que des constructions similaires existent en général. Par ailleurs, notre travaille est lié au treillis de ν-Tamari de Préville-Ratelle et Viennot.
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Submitted on : Monday, February 10, 2020 - 3:15:07 PM
Last modification on : Thursday, July 8, 2021 - 3:50:40 AM
Long-term archiving on: : Monday, May 11, 2020 - 2:42:55 PM

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  • HAL Id : hal-02473090, version 1

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Cesar Ceballos, Viviane Pons. The s-weak order and s-permutahedra. 31st International Conference on "Formal Power Series and Algebraic Combinatorics", Jul 2019, Ljubljana, Slovenia. ⟨hal-02473090⟩

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