Research in Algebraic Combinatorics

该项目有两个相互关联的部分：(1)对称函数理论在置换枚举中的应用；(2)拓扑组合学的研究。在置换枚举领域，PI将继续研究置换共轭类上的主指数和超越数的联合分布，这是她与Shareshian在基于交换代数考虑的部分有序集的拓扑上的工作中开始的。主要指数和卓越数是麦克马洪在上世纪初提出的四种基本排列统计中的两种。虽然文献中对四种基本排列统计量的其他组合的联合分布进行了广泛的研究，但令人惊讶的是，直到PI和Shareshian的工作才考虑到主要指数和超越数的联合分布。本文证明了这种联合分布具有许多显著的性质，其中之一是欧拉多项式指数生成函数经典公式的一个新的q-类比。为了证明这个公式，PI和Shareshian引入了一类有趣的对称函数，称为圈型欧拉拟对称函数，它改进了在各种表示理论和枚举环境中出现的对称函数，例如Askey和Ismail对多集无序的MacMahon枚举数的改进，Procesi和Stanley关于Coxeter复形的环变数的工作，以及Stanley关于对称色多项式的工作。圈型q-欧拉多项式和圈型欧拉拟对称函数似乎具有许多有趣的性质，但尚未得到证明。项目的第二部分包括对偏序集拓扑研究的延续，从而导致了主要指标和超越数联合分布的工作。由Bjorner和Welker引入的一种称为Rees积的偏序集运算，作为交换代数中Rees构造的组合模拟，是这项工作的核心，同样重要的还有偏序集的q-模拟概念。虽然PI和Shareshian的工作代表了置换枚举的新发展，但更有趣的是，它起源于置换枚举之外的工作，即拓扑组合学、交换代数、代数几何中的环变。在更深层次上理解所有这些联系的基础是一个有趣的问题。这些更深层次的基础可能有助于解释关于主要指数和卓越数联合分布的新结果的不断发现。该基金支持的研究领域是代数组合学，这是一个数学领域，旨在发展组合学（计算、排列和分析具体离散构型的科学）与涉及复杂抽象代数结构的纯数学领域之间的联系。我们的想法是利用这些联系来获得更深入的见解，并解决组合学和其他领域的问题。组合学中研究的离散构型出现在数学、计算机科学、物理学、生物学和工程学的各个领域；DNA序列、系统发育树和通信网络都是离散配置的例子。组合方法在这些领域中发挥着越来越重要的作用。

There are two interconnected parts to the proposed project: (1)the application of symmetric function theory to permutation enumeration and (2)research in topological combinatorics. In the area of permutation enumeration, the PI will continue a study of the joint distribution of the major index and the excedance number on conjugacy classes of permutations, which was initiated in her work with Shareshian on the topology of a partially ordered set arising from commutative algebra considerations. The major index and excedance number are two of four fundamental permutation statistics introduced by MacMahon at the beginning of the last century. Although the joint distribution of other combinations of the four fundamental permutation statistics have been extensively studied in the literature, surprisingly the joint distribution of the major index and excedance number had not been considered until the work of the PI and Shareshian. In this work it is shown that this joint distribution has many remarkable properties, one of which is a new q-analog of the classical formula for the exponential generating function of the Eulerian polynomials. In order to prove this formula, the PI and Shareshian introduced an intriguing class of symmetric functions called cycle-type Eulerian quasisymmetric functions, which refine symmetric functions that have occurred in various representation theoretic and enumerative contexts such as in Askey and Ismail's refinement of MacMahon's enumerator of multiset derangements, in work of Procesi and Stanley on toric varieties of Coxeter complexes, and in Stanley's work on symmetric chromatic polynomials. There are many interesting properties that the cycle type q-Eulerian polynomials and the cycle-type Eulerian quasisymmetric functions seem to possess, but have yet to be proved. The second part of the project includes a continuation of the research in poset topology that led to the work on the joint distribution of major index and excedance number. A poset operation, called Rees product introduced by Bjorner and Welker as a combinatorial analog of the Rees construction in commutative algebra, is central to this work, as is the notion of q-analog of a poset. While the work of the PI and Shareshian represents a novel development in permutation enumeration, what makes it even more interesting is that it has arisen from, and has found intimate connections to work outside of permutation enumeration i.e., topological combinatorics, commutative algebra, toric varieties in algebraic geometry. It is an intriguing problem to understand at a deeper level the basis for all these connections. These deeper underpinnings may help to explain the ongoing discovery of new results on the joint distribution of major index and excedance number.The research supported by this grant is in algebraic combinatorics, which is an area of mathematics that seeks to develop connections between combinatorics (the science of counting, arranging and analyzing concrete discrete configurations) and fields of pure mathematics that involve sophisticated abstract algebraic structures. The idea is to use these connections to gain deeper insights and solve problems in combinatorics and in the other fields. The discrete configurations that are studied in combinatorics arise in various fields of mathematics, computer science, physics, biology and engineering; DNA sequences, phylogenetic trees, and communications networks are all examples of discrete configurations. Combinatorial methods are playing an increasing role in these fields.