Séminaire Lotharingien de Combinatoire, 93B.121 (2025), 12 pp.
Eva Philippe and Vincent Pilaud
The s-Permutahedron and its Lattice Quotients
Abstract.
For a tuple s of non-negative integers, the s-weak order is a lattice on s-trees, generalizing the weak order on permutations.
We describe its join irreducible elements, its canonical join representations, and its forcing order in terms of combinatorial objects, generalizing the arcs, non-crossing arc diagrams, and subarc order for the weak order.
We then extend the theory of shards and shard polytopes to construct geometric realizations of the s-weak order and all its lattice quotients as polyhedral complexes, generalizing the quotient fans and quotientopes of the weak order.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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