Hyperplane arrangements with a lattice of regions
Discrete & Computational Geometry
Some properties of crossings and partitions
Discrete Mathematics
Non-crossing partitions for classical reflection groups
Discrete Mathematics
Subdivisions and triangulations of polytopes
Handbook of discrete and computational geometry
The order dimension of the poset of regions in a hyperplane arrangement
Journal of Combinatorial Theory Series A
Noncrossing Partitions for the Group Dn
SIAM Journal on Discrete Mathematics
Lattice congruences, fans and Hopf algebras
Journal of Combinatorial Theory Series A
Chains in the Noncrossing Partition Lattice
SIAM Journal on Discrete Mathematics
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We define a new lattice structure $(W,\preceq)$ on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC驴(W) as a sublattice. The new construction of NC驴(W) yields a new proof that NC驴(W) is a lattice. The shard intersection order is graded and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of $(W,\preceq)$ , like Möbius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of NC驴(W). There is a natural dimension-preserving bijection between simplices in the order complex of $(W,\preceq)$ (i.e. chains in $(W,\preceq)$ ) and simplices in a certain pulling triangulation of the W-permutohedron. Restricting the bijection to the order complex of NC驴(W) yields a bijection to simplices in a pulling triangulation of the W-associahedron.The lattice $(W,\preceq)$ is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of W. Indeed, most of the results of the paper are proven in the more general setting of simplicial hyperplane arrangements.