  We introduce and study a family of simplicial complexes associated to an
arbitrary finite root system and a nonnegative integer parameter m. For m=1,
our construction specializes to the (simplicial) generalized associahedra or,
equivalently, to the cluster complexes for the cluster algebras of finite type.
  Our computation of the face numbers and h-vectors of these complexes produces
the enumerative invariants defined in other contexts by C.A.Athanasiadis,
suggesting links to a host of well studied problems in algebraic combinatorics
of finite Coxeter groups, root systems, and hyperplane arrangements.
  Recurrences satisfied by the face numbers of our complexes lead to
combinatorial algorithms for determining Coxeter-theoretic invariants. That is,
starting with a Coxeter diagram of a finite Coxeter group, one can compute the
Coxeter number, the exponents, and other classical invariants by a recursive
procedure that only uses most basic graph-theoretic concepts applied to the
input diagram.
  In types A and B, we rediscover the constructions and results obtained by
E.Tzanaki <math.CO/0501100>.
