Standard fillings to parking functions

Elizabeth Niese

doi:10.46298/dmtcs.3075

Abstract

The Hilbert series of the Garsia-Haiman module can be written as a generating function of standard fillings of Ferrers diagrams. It is conjectured by Haglund and Loehr that the Hilbert series of the diagonal harmonics can be written as a generating function of parking functions. In this paper we present a weight-preserving injection from standard fillings to parking functions for certain cases. La série Hilbert du module Garsia-Haiman peut être écrite comme fonction génératrice de tableaux des diagrammes Ferrers. Haglund et Loehr conjecturent que la série Hilbert de l'harmonic diagonale peut être écrite comme fonction génératrice des fonctions parking. Dans cet essai nous présentons une injection des tableaux vers les fonctions parking pour certains cas.

Highlights

Over the past twenty years, Macdonald polynomials have become a central object of study in the theory of symmetric functions
In [Mac88], Macdonald introduces the dual basis, Qμ, and the integral basis, Jμ, which can both be computed from Pμ by multiplication by a suitable rational expression that depends on μ
There is a bijection between parking functions and labeled Dyck paths [Loe05, Loe11]

Summary

Introduction

Over the past twenty years, Macdonald polynomials have become a central object of study in the theory of symmetric functions. There is a bijection between parking functions and labeled Dyck paths [Loe, Loe11]. For a parking function f ∈ Pn corresponding to the labeled Dyck path P , set area(f ) = area(P ), g(f ) = g(P ), and p(f ) = p(P ).

Weight-preserving injection

Injection when μ is a hook shape

Conclusion