Combinatorial Structures for Permutation Enumeration and Diagonal Harmonic Modules

PI计划在三个不同领域进行研究。首先，PI计划扩展Brenti，Remmel，Beck，兰利，瓦格纳和其他人的工作，他们使用了从对称函数环到多项式环F[x]的某些同态，对于适当选择的域F，找到新的生成函数用于置换统计。最近发现的方法由PI和他的学生产生了许多新的应用这些方法允许一个找到生成函数为321避免排列，交替排列，和排列枚举统计的各种子群的花圈产品的有限群与对称群。PI计划将这些方法系统化并加以推广。其次，PI计划研究对角调和环的各种子模块的组合学，这与麦克唐纳多项式理论密切相关。Garsia，Haiman，Haglund，Loehr和PI的工作导致了希尔伯特级数和生成函数的组合解释，这些生成函数用于交替表示在对角作用下的子模块的特征中，根据某些统计数据计算各类标记的Dyck路径。最近，Haiman，Haglund，Loehr，Remmel和Ulyanov给出了对角调和环的Frobenius级数中单项对称函数的系数的组合解释。PI在这一领域的研究目标是证明其中的一些假设，并给出对角调和环的Frobenius级数的完整组合解释。最后，PI还计划工作的（p，q）-类似物的车理论和各种扩展的车理论和理论的枚举的生成森林的各种有向图研究Egecioglu，威廉姆森和PI。这个项目涉及的各种问题出现在研究代数组合。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，格路径，生成树和非附加车在广义棋盘上的位置，在帮助我们理解某些代数结构方面发挥着重要作用。相反，在代数组合学中，人们希望了解各种代数结构如何为这些经典组合结构的理论提供新的见解。在这个项目中提出的三个主要研究领域，即排列枚举理论、对角调和环理论和车理论都是活跃的研究领域，代数和组合方法之间有着美丽的相互作用。

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.