Combinatorial Structures for Permutation Enumeration and Diagonal Harmonic Modules

排列枚举和对角调和模的组合结构

• 批准号: 0400507
• 负责人: Jeffrey Remmel
• 金额: $ 10.5万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2007-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400507&HistoricalAwards=false
• 关键词: Combinatorial; Structures; Permutation; Enumeration; Diagonal

The PI plans to pursue research in three different areas. First the PI plans to extend the work of Brenti, Remmel, Beck, Langley, Wagner and others who have used certain homomorphisms from the ring of symmetric functions to a polynomial ring F[x], for an appropriately chosen field F, to find new generating functions for permutations statistics. Recently discovered methods by the PI and his students have produced many new applications of these methods which allow one to find generating functions for 321-avoiding permutations, alternating permutations, and permutation enumeration statistics for various subgroups of the wreath product of a finite group with the symmetric group. The PI plans to systemize and extend such methods. Second, the PI plans to study the combinatorics of various submodules of the ring of diagonal harmonics which is intimately connected with the theory of Macdonald polynomials. The work of Garsia, Haiman, Haglund, Loehr and the PI have lead to combinatorial interpretations of the Hilbert series and the generating function for the occurrences of the alternating representation in the character of such submodules under the diagonal action in terms of counting various classes of labeled Dyck paths according to certain statistics. More recently, Haiman, Haglund, Loehr, Remmel and Ulyanov have given a conjectured combinatorial interpretation of the coefficient of a monomial symmetric function in the Frobenius series of the ring of diagonal harmonics. The goal of the PI's research in this area is to prove some of these conjectures and to give a full combinatorial interpretation of the Frobenius series of the ring of diagonal harmonics. Finally the PI also plans to work on (p,q)-analogues of rook theory and various extension of rook theory and the theory of enumeration of spanning forests of various directed graphs studied by Egecioglu, Williamson and the PI.This project is concerned with various problems which arise in the study of algebraic combinatorics. In particular, the goal of this project is to increase our understanding of how various classical combinatorial structures such as permutations, lattice paths, spanning trees and placements of non-attaching rooks on generalized chess boards play a fundamental role in helping us understand certain algebraic structures. Conversely, in algebraic combinatorics, one wants to understand how various algebraic structures can give new insights into the theory of these classical combinatorial structures. The three main areas of research proposed in this project, namely the theory of permutation enumeration, the theory of the ring of diagaonal harmonics and rook theory are all active areas of research were there is a beautiful interplay between algebraic and combinatorial methods.

PI计划在三个不同的领域进行研究。首先，PI计划将Brenti，Remmel，Beck，Langley，Wagner等人的工作从对称函数环扩展到多项式环F[x]，对于适当选择的域F，找到新的置换统计量的生成函数。PI和他的学生最近发现的方法已经产生了这些方法的许多新的应用，使得人们可以找到321的母函数-避免排列、交替排列和对称群的花圈积的各种子群的排列计数统计。国际和平研究所计划将这些方法系统化并加以推广。其次，PI计划研究与Macdonald多项式理论密切相关的对角调和环的各个子模的组合学。Garsia，Haiman，Haglund，Loehr和PI的工作得到了Hilbert级数的组合解释，以及在对角线作用下，根据某种统计对各类标记Dyck路进行计数，这些子模的特征的交替表示出现的母函数。最近，Haiman，Haglund，Loehr，Remmel和Ulyanov对对角调和环的Frobenius级数中的单项对称函数的系数给出了猜想的组合解释。PI在这一领域的研究目标是证明其中一些猜想，并对对角调和环的Frobenius级数给出完整的组合解释。最后，PI还计划研究Rook理论的(p，q)-类似物以及由Egecioglu、Williamson和PI研究的各种有向图的生成林计数理论和Rook理论的各种扩展。这个项目涉及代数组合学研究中出现的各种问题。特别是，这个项目的目标是增加我们对各种经典组合结构的理解，如排列、格路、生成树和非附着棋子在广义棋盘上的放置如何在帮助我们理解某些代数结构方面发挥基础作用。相反，在代数组合学中，人们想要了解不同的代数结构如何给这些经典组合结构的理论带来新的见解。这个项目提出的三个主要研究领域，即置换计数理论、对角调和环理论和Rook理论都是活跃的研究领域，因为代数方法和组合方法之间存在着美丽的相互作用。