Alexander Duality and Rational Associahedra

A recent pair of papers of Armstrong, Loehr, and Warrington and Armstrong, Williams, and the author initiated the systematic study of {\em rational Catalan combinatorics} which is a generalization of Fuss-Catalan combinatorics (which is in turn a generalization of classical Catalan combinatorics). The latter paper gave two possible models for a rational analog of the associahedron which attach simplicial complexes to any pair of coprime positive integers a < b. These complexes coincide up to the Fuss-Catalan level of generality, but in general one may be a strict subcomplex of the other. Verifying a conjecture of Armstrong, Williams, and the author, we prove that these complexes agree up to homotopy and, in fact, that one complex collapses onto the other. This reconciles the two competing models for rational associahedra. As a corollary, we get that the involution (a < b) \longleftrightarrow (b-a < b) on pairs of coprime positive integers manifests itself topologically as Alexander duality of rational associahedra. This collapsing and Alexander duality are new features of rational Catalan combinatorics which are invisible at the Fuss-Catalan level of generality.Comment: 23 page

Project uncertainty, project risk and project leadership : a policy capturing study of New Zealand project managers : a thesis presented in partial fulfillment of the requirements for the degree of Master of Arts in Psychology at Massey University, Wellington, New Zealand

Cooperation between project practice and project research could help reduce failure rates for projects in New Zealand and globally. The current research used a “policy capturing” method - systematically varying sources of project uncertainty (policy cues) to explore project leadership responses. A contingency model proposed that project uncertainty (low path-goal clarity, low team cohesion, and high technical complexity) would lead to greater perceptions of project risk (scope/quality, budget, schedule, and project team satisfaction) that would negatively predict the (rated) effectiveness of transactional leadership style and positively predict ratings for transformational style. In total, n=131 experienced project managers rated the effectiveness of leadership styles from ‘not effective’ to ‘extremely effective’. Greater uncertainty produced higher perceived risks that reduced the rated effectiveness of transactional leadership. Path-goal clarity was of particular importance as a policy cue, directly predicting transactional leadership ratings (R=-0.189). These results are consistent with the task-orientation of traditional project management. However, the results for transformational style were unexpected - only team cohesion predicted transformational leadership ratings (negatively) (R= -0.119) and no link between risk and transformational leadership was found. Possible reasons for the ‘disconnect between transformational leadership, uncertainty and risk are discussed and further research suggested

In Defense of Trump's Jerusalem Move

The Texas Orato

Cyclic sieving and cluster multicomplexes

Reiner, Stanton, and White \cite{RSWCSP} proved results regarding the enumeration of polygon dissections up to rotational symmetry. Eu and Fu \cite{EuFu} generalized these results to Cartan-Killing types other than A by means of actions of deformed Coxeter elements on cluster complexes of Fomin and Zelevinsky \cite{FZY}. The Reiner-Stanton-White and Eu-Fu results were proven using direct counting arguments. We give representation theoretic proofs of closely related results using the notion of noncrossing and semi-noncrossing tableaux due to Pylyavskyy \cite{PN} as well as some geometric realizations of finite type cluster algebras due to Fomin and Zelevinsky \cite{FZClusterII}.Comment: To appear in Adv. Appl. Mat

Enumeration of connected Catalan objects by type

Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted plane trees are four classes of Catalan objects which carry a notion of type. There exists a product formula which enumerates these objects according to type. We define a notion of `connectivity' for these objects and prove an analogous product formula which counts connected objects by type. Our proof of this product formula is combinatorial and bijective. We extend this to a product formula which counts objects with a fixed type and number of connected components. We relate our product formulas to symmetric functions arising from parking functions. We close by presenting an alternative proof of our product formulas communicated to us by Christian Krattenthaler which uses generating functions and Lagrange inversion