Combinatorial Structures for Permutation Enumeration and Diagonal Harmonic Modules

PI计划在三个不同的领域进行研究。首先，PI计划扩展Brenti， Remmel, Beck, Langley， Wagner等人的工作，他们使用对称函数环的某些同态到多项式环F[x]，对于适当选择的域F，寻找新的置换统计生成函数。PI和他的学生最近发现的方法产生了这些方法的许多新应用，这些方法允许人们找到321-避免置换，交替置换和有限群与对称群的环积的各种子群的置换枚举统计的生成函数。PI计划将这些方法系统化并加以扩展。其次，PI计划研究与麦克唐纳多项式理论密切相关的对角谐波环的各种子模的组合。Garsia, Haiman, Haglund， Loehr和PI的工作导致了Hilbert级数的组合解释，以及对角线作用下这些子模块特征中交替表示出现的生成函数，根据一定的统计计数各种标记的Dyck路径。最近，Haiman, Haglund, Loehr， Remmel和Ulyanov对对角谐波环的Frobenius级数中单项式对称函数的系数给出了一个猜想的组合解释。PI在这一领域的研究目标是证明其中的一些猜想，并给出对角谐波环的Frobenius级数的一个完整的组合解释。最后，PI还计划研究（p,q）——白嘴鸦理论的类似物和白嘴鸦理论的各种推广，以及Egecioglu、Williamson和PI研究的各种有向图的跨林枚举理论。本课题涉及代数组合学研究中出现的各种问题。特别地，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、晶格路径、生成树和非连接的车在广义棋盘上的位置如何在帮助我们理解某些代数结构方面发挥基本作用。相反，在代数组合学中，人们想要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。本项目提出的三个主要研究领域，即置换枚举理论、对角谐波环理论和白车理论，都是代数和组合方法之间美丽相互作用的活跃研究领域。

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.