Ph.D. (2006) Cornell University
Abstract: This thesis serves two purposes: it is a comprehensive introduction to the “Catalan combinatorics” of finite Coxeter groups, suitable for nonexperts, and it also introduces and studies a new generalization of the poset of noncrossing partitions. This poset is part of a “Fuss-Catalan combinatorics” of finite Coxeter groups, generalizing the Catalan combinatorics.
Our central contribution is the definition of a generalization NC(k)(W) of the poset of noncrossing partitions corresponding to each finite Coxeter group W and positive integer k. This poset has elements counted by a generalized Fuss-Catalan number \Cat(k)(W), defined in terms of the invariant degrees of W. We develop the theory of this poset in detail. In particular, we show that it is a graded semilattice with beautiful structural and enumerative properties. We count multichains and maximal chains in NC(k)(W). We show that the order complex of NC(k)(W) is shellable and hence Cohen-Macaulay, and we compute the reduced Euler characteristic of this complex. We show that the rank numbers of NC(k)(W) are polynomial in k; this defines a new family of polynomials (called Fuss-Narayana) associated to the pair (W,k). We observe some fascinating properties of these polynomials.
We study the structure NC(k)(W) more specifically when W is a classical type A or type B Coxeter group. In these cases, we show that NC(k)(W) is isomorphic to a poset of “noncrossing” set partitions in which each block has size divisible by k. Hence, we refer to NC(k)(W) in general as the poset of “ k-divisible noncrossing partitions.” In this case, we prove rank-selected and type-selected enumeration formulas for multichains in NC(k)(W). We also describe new bijections between multichains of classical noncrossing partitions and classical k-divisible noncrossing partitions.
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 complex of Fomin and Reading. We give a basic introduction to these topics and describe several new conjectures relating these three families of “Fuss-Catalan objects.” We mention connections with the theories of cyclic sieving and diagonal harmonics.