Symmetric Functions: Combinatorial Identities and Bijections

代数组合学是数学的一个分支，它利用代数和组合学之间的相互作用，使得一个领域的证明方法可以用来证明另一个领域的定理和定理。这个特别的项目集中在证明组合身份和双射对称函数通过利用组合对象，如tableaux和树。这项工作的动力来自投票理论中的问题，它建立在我以前的工作基础上。这项建议的重点是发展一个新的领域的对称函数理论，在投票理论的问题的动机。这项研究的核心是关于排名模式的问题。在对称函数理论中，什么具有挑战性和吸引力的问题是由排序模式问题激发的？这是一种独特的方法。这个建议不是专注于一种技术或一种类型的对称函数，而是针对一个共同的起点，并问，如果我们从这个点出发，会出现什么有趣的对称函数问题？从这个中心枢纽辐射出各种感兴趣的辐条：布尔乘积多项式，避免模式的置换模式，交替树。 什么是排名模式？这些是离散物体的有序排列。计算这种排列的数量并不困难：显然，所有可能排列的集合就是所有可能排列的集合;然而，现实比这更微妙，在许多应用中，并非所有排列都会发生。例如，您如何为用户在选举中对候选人的排名建模？一个流行的模型优先考虑某些排名，但这导致了一个新的问题：在模型中有多少排名是可能的？一般来说，对于更高维的模型来说，双射是很困难的，数学家们通过对组合对象进行双射来得出答案。通过最近的工作，这个模型和超平面安排之间的联系已经取得，这导致了涉及罗宾逊-申斯特算法，交替符号矩阵，交替树，和模式避免排列。 我的目标就是这些人。沿着的方式，我将证明结果涉及正式计数的组合对象和禁止配置的特征。大部分的研究在我的职业生涯都集中在对称函数身份的组合证明，并为此我增加了一些最近的结果截断排名选票选举的后果。这个社会选择的方向交叉影响了我的对称函数研究，并导致了新的想法和新的轨迹。虽然这个提议的目标是理论上的，但也有通往应用领域的桥梁，包括在某些投票系统模型中显示优势和局限性的潜力，我希望社会选择社区成为我的结果的重要额外受众。

Algebraic combinatorics is the branch of mathematics that exploits the interplay between algebra and combinatorics so that the methods of proof in one area can be employed to prove theorems and conjectures in the other area. This particular project concentrates on proving combinatorial identities and bijections for symmetric functions through the exploitation of combinatorial objects such as tableaux and trees. The impetus for this work comes from problems in voting theory and it builds on my previous work. This proposal focuses on the development of a new sphere of symmetric function theory, motivated by problems in voting theory. At the very heart of this research are questions about ranking patterns. What challenging and compelling questions in symmetric function theory are inspired by ranking pattern questions? This is a distinctive approach. Instead of focusing on a technique or a type of symmetric function, this proposal targets a common origin point and asks, if we journey out from this point, what interesting symmetric function questions arise? From this central hub radiate out the various spokes of interest: Boolean product polynomials, pattern-avoiding permutation patterns, alternating trees. What are ranking patterns? These are ordered arrangements of discrete objects. Counting the number of such arrangements is not difficult: obviously, the set of all possible rankings is the set of all possible permutations; however, reality is more subtle than that and, in many applications, not all rankings occur. For example, how do you model users' rankings of candidates in an election? One popular model prioritizes certain rankings, but this leads to a new problem: how many rankings are possible in the model? Answering in general can be difficult for higher dimensional models, and mathematicians have derived answers by forging bijections to combinatorial objects. Through recent work, connections between this model and hyperplane arrangements have been made, and this has led to conjectures involving the Robinson-Schensted algorithm, alternating sign matrices, alternating trees, and pattern-avoiding permutations.  It is these conjectures I will target. Along the way I will prove results involving formal counting of combinatorial objects and the characterization of forbidden configurations. The majority of the research throughout my career has concentrated on combinatorial proofs of symmetric function identities, and to this I have added some recent results on the ramifications of truncation on ranked ballot elections. This social choice direction has cross pollinated my symmetric function research, and led to new ideas and new trajectories. While the goals of this proposal are theoretical, there is this bridge to the applied realm as well, including, the potential to show advantages and limitations in certain voting system models, and I expect the social choice community to be an important additional audience for my results.