Research in Algebraic Combinatorics

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The PI continues her investigation of algebraic and topological aspects of simplicialcomplexes associated with partially ordered sets (posets) and monotone graph properties. The theory of poset topology provides a deep and fundamental link between combinatorics and other branches of mathematics such as topology, algebra and geometry. There are five parts to the project. The first three parts are connected with the study of topological properties of a new poset operation coming from commutative algebra, called Rees product. By studying the Rees product of two very simple posets, the PI and John Shareshian have discovered some remarkable enumerative and algebraic identities. The most striking of the enumerative identities is a conjectured q-analog of a well-known identity for the Eulerian polynomials in terms of the joint distribution of the major index and the excedance index. In Part 4, the PI proposes to obtain a k-analog of a well-known relationship between the homology of the partition lattice and the homology of the complex of graphs that are not connected. The PI and Shareshian have a precise conjecture on what that should be, involving the so called1 mod k partition poset and the complex of graphs that are not k-edge connected. In Part 5, the PI proposes to continue her study of the matching complex, the chessboard complex and variations. These complexes arise in diverse settings such as group theory, discrete geometry and commutative algebra. Algebraic combinatorics is an area of mathematics that seeks to establish connections between combinatorics and fields of pure mathematics that involve algebra. The idea is to use these connections to enrich combinatorics and the other fields. Combinatorics is the science of counting, arranging and analyzing discrete configurations. A communications network is an example of a fundamental discrete configuration called a graph. Graphs and other discrete configurations arise in various fields of mathematics, computer science, physics and biology. Combinatorial methods are playing an increasing role in these fields.