Drew Armstrong
Drew Armstrong

Ph.D. (2006) Cornell University

First Position
Dissertation
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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 “FussCatalan 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 FussCatalan 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 CohenMacaulay, 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 FussNarayana) 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 “ kdivisible noncrossing partitions.” In this case, we prove rankselected and typeselected enumeration formulas for multichains in NC^{(k)}(W). We also describe new bijections between multichains of classical noncrossing partitions and classical kdivisible 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 “FussCatalan objects.” We mention connections with the theories of cyclic sieving and diagonal harmonics.