Séminaire Lotharingien de Combinatoire, 82B.76 (2019), 12 pp.

Cesar Ceballos and Viviane Pons

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.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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