Journal of Combinatorics
Volume 13 (2022)
Poset topology of $s$ weak order via SB-labelings
Pages: 357 – 395
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.
$s$-weak order, $s$-Tamari lattice, SB-labeling, poset topology
2010 Mathematics Subject Classification
The author was supported by NSF grants DMS-1953931 and DMS-1500987.
Received 5 September 2020
Accepted 22 May 2021
Published 31 March 2022