This memoir constitutes the author's PhD thesis at Cornell University. It
serves both as an expository work and as a description of new research. At the
heart of the memoir, we introduce and study a poset NC(k)(W) for each
finite Coxeter group W and for each positive integer k. When k=1, our
definition coincides with the generalized noncrossing partitions introduced by
Brady-Watt and Bessis. When W is the symmetric group, we obtain the poset of
classical k-divisible noncrossing partitions, first studied by Edelman.
Along the way, we include a comprehensive introduction to related background
material. Before defining our generalization NC(k)(W), we develop from
scratch the theory of algebraic noncrossing partitions NC(W). This involves
studying a finite Coxeter group W with respect to its generating set T of
{\em all} reflections, instead of the usual Coxeter generating set S. This is
the first time that this material has appeared in one place.
Finally, it turns out that our poset NC(k)(W) shares many enumerative
features in common with the ``generalized nonnesting partitions'' of
Athanasiadis and the ``generalized cluster complexes'' of Fomin and Reading. In
particular, there is a generalized ``Fuss-Catalan number'', with a nice closed
formula in terms of the invariant degrees of W, that plays an important role
in each case. We give a basic introduction to these topics, and we describe
several conjectures relating these three families of ``Fuss-Catalan objects''.Comment: Final version -- to appear in Memoirs of the American Mathematical
Society. Many small improvements in exposition, especially in Sections 2.2,
4.1 and 5.2.1. Section 5.1.5 deleted. New references to recent wor
