Let A be any subspace arrangement in R n defined over the integers and let F q denote the finite field with q elements. Let q be a large prime. We prove that the characteristic polynomial /(A, q) of A counts the number of points in F n q that do not lie in any of the subspaces of A, viewed as subsets of F n q . This observation, which generalizes a theorem of Blass and Sagan about subarrangements of the B n arrangement, reduces the computation of /(A, q) to a counting problem and provides an explanation for the wealth of combinatorial results discovered in the theory of hyperplane arrangements in recent years. The basic idea has its origins in the work of Crapo and Rota (1970). We find new classes of hyperplane arrangements whose characteristic polynomials have simple form and very often factor completely over the nonnegative integers.

For an irreducible, crystallographic root system Φ in a Euclidean space V and a positive integer m, the arrangement of hyperplanes in V given by the affine equations (α, x) = k, for α ∈ Φ and k = 0, 1, . . . , m, is denoted here by A m Φ . The characteristic polynomial of A m Φ is related in the paper to that of the Coxeter arrangement A Φ (corresponding to m = 0), and the number of regions into which the fundamental chamber of A Φ is dissected by the hyperplanes of A m Φ is deduced to be equal to the product i=1 (e i + mh + 1)/(e i + 1), where e 1 , e 2 , . . . , e are the exponents of Φ and h is the Coxeter number. A similar formula for the number of bounded regions follows. Applications to the enumeration of antichains in the root poset of Φ are included.

Abstract. The poset of noncrossing partitions can be naturally defined for any finite Coxeter group W . It is a self-dual, graded lattice which reduces to the classical lattice of noncrossing partitions of {1, 2, . . . , n} defined by Kreweras in 1972 when W is the symmetric group Sn, and to its type B analogue defined by the second author in 1997 when W is the hyperoctahedral group. We give a combinatorial description of this lattice in terms of noncrossing planar graphs in the case of the Coxeter group of type Dn, thus answering a question of Bessis. Using this description, we compute a number of fundamental enumerative invariants of this lattice, such as the rank sizes, number of maximal chains, and Möbius function.We also extend to the type D case the statement that noncrossing partitions are equidistributed to nonnesting partitions by block sizes, previously known for types A, B, and C. This leads to a (caseby-case) proof of a theorem valid for all root systems: the noncrossing and nonnesting subspaces within the intersection lattice of the Coxeter hyperplane arrangement have the same distribution according to W -orbits.

Abstract. Let Φ be an irreducible crystallographic root system with Weyl group W , coroot latticeQ and Coxeter number h, spanning a Euclidean space V , and let m be a positive integer. It is known that the set of regions into which the fundamental chamber of W is dissected by the hyperplanes in V of the form (α, x) = k for α ∈ Φ and k = 1, 2, . . . , m is equinumerous to the set of orbits of the action of W on the quotientQ/ (mh + 1)Q. A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of Φ, are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case m = 1 and Φ = A n−1 .

Abstract. We give a case-free proof that the lattice of noncrossing partitions associated to any finite real reflection group is EL-shellable. Shellability of these lattices was open for the groups of type Dn and those of exceptional type and rank at least three.