Research in Algebraic Combinatorics

PI继续研究与偏序集(偏序集)和单调图性质有关的简单复形的代数和拓扑方面。偏序集拓扑理论在组合数学和其他数学分支如拓扑学、代数和几何学之间提供了一种深刻而基本的联系。该项目包括五个部分。前三部分是关于交换代数中一种新的偏序集运算的拓扑性质的研究，称为Rees积。通过研究两个非常简单的偏序集的Rees积，Pi和John Shareshian发现了一些值得注意的计数恒等式和代数恒等式。计数恒等式中最引人注目的是关于主要指标和超越指标的联合分布的欧拉多项式的一个众所周知的恒等式的猜想Q模拟。在第四部分中，PI建议得到划分格的同调和不连通的图的复图的同调之间的已知关系的k-模拟。PI和Shareshian对这应该是什么有一个精确的猜测，涉及所谓的1mod k划分偏序集和不是k边连接的图的复数。在第五部分中，PI建议继续她对匹配情结、棋盘情结和变体的研究。这些复合体出现在不同的环境中，如群论、离散几何和交换代数。代数组合学是一个数学领域，它试图在组合学和涉及代数的纯数学领域之间建立联系。我们的想法是利用这些联系来丰富组合学和其他领域。组合学是计算、排列和分析离散构型的科学。通信网络是称为图的基本离散配置的一个例子。图形和其他离散构型出现在数学、计算机科学、物理和生物学的各个领域。组合方法在这些领域发挥着越来越大的作用。

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.