Geodesic
The \(s\)-weak order and \(s\)-permutahedra. II: The combinatorial complex of pure intervals
The electronic journal of combinatorics, Tome 31 (2024) no. 3
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This paper introduces the geometric foundations for the study of the $s$-permutahedron and the $s$-associahedron, two objects that encode the underlying geometric structure of the $s$-weak order and the $s$-Tamari lattice. We introduce the $s$-permutahedron as the complex of pure intervals of the $s$-weak order, present enumerative results about its number of faces, and prove that it is a combinatorial complex. This leads, in particular, to an explicit combinatorial description of the intersection of two faces. We also introduce the $s$-associahedron as the complex of pure $s$-Tamari intervals of the $s$-Tamari lattice, show some enumerative results, and prove that it is isomorphic to a well chosen $\nu$-associahedron. Finally, we present three polytopality conjectures, evidence supporting them, and some hints about potential generalizations to other finite Coxeter groups.
DOI : 10.37236/12438
Classification : 06A07, 06B05, 20F55, 52B05, 06B10
Mots-clés : weak order, pure intervals, \(s\)-permutahedron, \(s\)-associahedron, Tamari lattice

Cesar Ceballos  1   ; Viviane Pons 

1 TU Graz 
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     title = {The \(s\)-weak order and \(s\)-permutahedra. {II:} {The} combinatorial complex of pure intervals},
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Cesar Ceballos; Viviane Pons. The \(s\)-weak order and \(s\)-permutahedra. II: The combinatorial complex of pure intervals. The electronic journal of combinatorics, Tome 31 (2024) no. 3. doi: 10.37236/12438

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