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Séminaire Lotharingien de Combinatoire, 84B.28 (2020), 12 pp.

# Stephen Lacina

# Poset Topology of *s*-Weak Order via SB-Labelings

**Abstract.**
Ceballos and Pons generalized weak order on permutations to a partial order on certain labeled trees, thereby introducing a new class of lattices called *s*-weak order. They also generalized the Tamari lattice by defining a particular sublattice of *s*-weak order called the *s*-Tamari lattice. We prove that the homotopy type of each open interval in *s*-weak order and in the *s*-Tamari lattice is either a ball or sphere. We do this by giving *s*-weak order and the *s*-Tamari lattice a type of edge labeling known as an SB-labeling. We characterize which intervals are homotopy equivalent to spheres and which are homotopy equivalent to balls; we also determine the dimension of the spheres for the intervals yielding spheres.

Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.

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