ABSTRACT. We derive two multivariate generating functions for threedimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite abelian subgroup G of SO(3). These generating functions turn out to be orbifold Donaldson-Thomas partition functions for the orbifold [C 3 /G]. We need only the vertex operator methods of Okounkov-ReshetikhinVafa for the easy case G = Z n ; to handle the considerably more difficult case G = Z 2 × Z 2 , we will also use a refinement of the author's recent q-enumeration of pyramid partitions.In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold [C 3 /G]. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the Hard Lefschetz condition.

We study random domino tilings of the Aztec diamond with different weights for horizontal and vertical dominoes. A domino tiling of an Aztec diamond can also be described by a particle system which is a determinantal process. We give a relation between the correlation kernel for this process and the inverse Kasteleyn matrix of the Aztec diamond. This gives a formula for the inverse Kasteleyn matrix which generalizes a result of Helfgott. As an application, we investigate the asymptotics of the process formed by the southern dominoes close to the frozen boundary. We find that at the northern boundary, the southern domino process converges to a thinned Airy point process. At the southern boundary, the process of holes of the southern domino process converges to a multiple point process that we call the thickened Airy point process. We also study the convergence of the domino process in the unfrozen region to the limiting Gibbs measure.

The inverse Kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics. In this paper, we consider three different weightings of domino tilings of the Aztec diamond and show using recurrence relations, we can compute the inverse Kasteleyn matrix. These weights are the one-periodic weighting where the horizontal edges have one weight and the vertical edges have another weight, the q vol weighting which corresponds to multiplying the product of tile weights by q if we add a 'box' to the height function and the two-periodic weighting which exhibits a flat region with defects in the center. col 36 5.3. Proof of Theorem 5.1 37 6. Two-Periodic Weighting 42

The Systematic Screening for Behavior Disorders (SSBD), a multistage screening system designed to identify elementary school—age children at risk for emotional and behavioral disorders, was evaluated for use with middle and junior high school students. During SSBD Stage 1, teachers identified 123 students in grades 6 through 9 with characteristics of internalizing and externalizing disorders. Teachers then completed SSBD Stage 2 behavior rating scales, the Teacher Report Form, and the Social Skill Rating System on 119 of these students identified as at-risk during Stage 1. Office discipline referrals and cumulative grade point averages for at-risk students were compared to those of students not designated by teachers. SSBD Stage 2 scores were compared with scores from the Teacher Report Form and Social Skill Rating System. Internal consistency and interrater reliability of the SSBD were also examined. Results provide evidence for the reliability and validity of SSBD ratings of early adolescent students.

Abstract. We define the equivariant Kazhdan-Lusztig polynomial of a matroid equipped with a group of symmetries, generalizing the nonequivariant case. We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank.