Skip to Main content Skip to Navigation
Conference papers

Special Cases of the Parking Functions Conjecture and Upper-Triangular Matrices

Abstract : We examine the $q=1$ and $t=0$ special cases of the parking functions conjecture. The parking functions conjecture states that the Hilbert series for the space of diagonal harmonics is equal to the bivariate generating function of $area$ and $dinv$ over the set of parking functions. Haglund recently proved that the Hilbert series for the space of diagonal harmonics is equal to a bivariate generating function over the set of Tesler matrices–upper-triangular matrices with every hook sum equal to one. We give a combinatorial interpretation of the Haglund generating function at $q=1$ and prove the corresponding case of the parking functions conjecture (first proven by Garsia and Haiman). We also discuss a possible proof of the $t = 0$ case consistent with this combinatorial interpretation. We conclude by briefly discussing possible refinements of the parking functions conjecture arising from this research and point of view. $\textbf{Note added in proof}$: We have since found such a proof of the $t = 0$ case and conjectured more detailed refinements. This research will most likely be presented in full in a forthcoming article.
Complete list of metadata

Cited literature [5 references]  Display  Hide  Download

https://hal.inria.fr/hal-01215036
Contributor : Coordination Episciences Iam <>
Submitted on : Tuesday, October 13, 2015 - 3:05:21 PM
Last modification on : Friday, June 28, 2019 - 2:48:03 PM
Long-term archiving on: : Thursday, April 27, 2017 - 12:17:03 AM

File

dmAO0156.pdf
Publisher files allowed on an open archive

Identifiers

  • HAL Id : hal-01215036, version 1

Collections

Citation

Paul Levande. Special Cases of the Parking Functions Conjecture and Upper-Triangular Matrices. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.635-644. ⟨hal-01215036⟩

Share

Metrics

Record views

93

Files downloads

583