The Partition Lattice in Many Guises

This dissertation is divided into four chapters. In Chapter 2 the equivariant homology groups of upper order ideals in the partition lattice are computed. The homology groups of these filters are written in terms of border strip Specht modules as well as in terms of links in an associated complex in the lattice of compositions. The classification is used to reproduce topological calculations of many well-studied subcomplexes of the partition lattice, including the d-divisible partition lattice and the Frobenius complex. In Chapter 3 the box polynomial B_{m,n}(x) is defined in terms of all integer partitions that fit in an m by n box. The real roots of the box polynomial are completely characterized, and an asymptotically tight bound on the norms of the complex roots is also given. An equivalent definition of the box polynomial is given via applications of the finite difference operator Delta to the monomial x^{m+n}. The box polynomials are also used to find identities counting set partitions with all even or odd blocks, respectively. Chapter 4 extends results from Chapter 3 to give combinatorial proofs for the ordinary generating function for set partitions with all even or all odd block sizes, respectively. This is achieved by looking at a multivariable generating function analog of the Stirling numbers of the second kind using restricted growth words. Chapter 5 introduces a colored variant of the ordered partition lattice, denoted Q_n^{\alpha}, as well an associated complex known as the alpha-colored permutahedron, whose face poset is Q_n^\alpha. Connections between the Eulerian polynomials and Stirling numbers of the second kind are developed via the fibers of a map from Q_n^{\alpha} to the symmetric group on n-element

A face iterator for polyhedra and more general finite locally branched lattices

We discuss a new memory-efficient depth-first algorithm and its implementation that iterates over all elements of a finite locally branched lattices. This algorithm can be applied to face lattices of polyhedra and various generalizations such as finite polyhedral complexes and subdivisions of manifolds, extended tight spans and closed sets of matroids. Its practical implementation is very fast compared to state-of-the-art implementations of previously considered algorithms.Comment: 13 pages including long examples and computational dat

Volume Segmentation of 3-dimensional Images

We present a practical method to segment large medical images that takes the whole 3-dimensional structure into account. We use a Union-Find data structure to record and maintain the necessary information during the segmentation process. Due to the large data size, we are forced to divide our process in two parts: a "weak segmentation" of the individual sections and a global integration of all the data. This method shows good results on computer tomographies

Shard polytopes

For any lattice congruence of the weak order on permutations, N. Reading proved that gluing together the cones of the braid fan that belong to the same congruence class defines a complete fan, called a quotient fan, and V. Pilaud and F. Santos showed that it is the normal fan of a polytope, called a quotientope. In this paper, we provide a simpler approach to realize quotient fans based on Minkowski sums of elementary polytopes, called shard polytopes, which have remarkable combinatorial and geometric properties. In contrast to the original construction of quotientopes, this Minkowski sum approach extends to type $B$.Comment: 73 pages, 35 figures; Version 2: minor corrections for final versio

Combinatorial Potpourri: Permutations, Products, Posets, and Pfaffians

In this dissertation we first examine the descent set polynomial, which is defined in terms of the descent set statistics of the symmetric group. Algebraic and topological tools are used to explain why large classes of cyclotomic polynomials are factors of the descent set polynomial. Next the diamond product of two Eulerian posets is studied, particularly by examining the effect this product has on their cd-indices. A combinatorial interpretation involving weighted lattice paths is introduced to describe the outcome of applying the diamond product operator to two cd-monomials. Then the cd-index is defined for infinite posets, with the calculation of the cd-index of the universal Coxeter group under the Bruhat order as an example. Finally, an extension of the Pfaffian of a skew-symmetric function, called the hyperpfaffian, is given in terms of a signed sum over partitions of n elements into blocks of equal size. Using a sign-reversing involution on a set of weighted, oriented partitions, we prove an extension of Torelli\u27s Pfaffian identity that results from applying the hyperpfaffian to a skew-symmetric polynomial

The Role of Synthetic Geometry in Representational Measurement Theory

Geometric representations of data and the formulation of quantitative models of observed phenomena are of main interest in all kinds of empirical sciences. To support the formulation of quantitative models, {\it representational measurement theory} studies the foundations of measurement. By mathematical methods it is analysed under which conditions attributes have numerical measurements and which numerical manipulations of the measurement values are meaningful (see Krantz et al.~(1971)). In this paper, we suggest to discuss within the measurement theory approach both, the idea of geometric representations of data and the request to provide algebraic descriptions of dependencies of attributes. We show that, within such a broader paradigm of representational measurement theory, synthetic geometry can play a twofold role which enriches the theory and the possibilities of data interpretation