Poset topology of $s$ weak order via SB-labelings

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.

Keywords

$s$-weak order, $s$-Tamari lattice, SB-labeling, poset topology

2010 Mathematics Subject Classification

06A07

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The author was supported by NSF grants DMS-1953931 and DMS-1500987.

Received 5 September 2020

Accepted 22 May 2021

Published 31 March 2022