Combinatorial Structures for Permutation Enumeration and Diagonal Harmonic Modules

University Of California-San Diego, La Jolla CA

Investigators

Abstract

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.