A Parking Function Setting for Nabla Images of Schur Functions

Abstract

In this article, we show how the compositional refinement of the ``Shuffle Conjecture'' due to Jim Haglund, Jennifer Morse, and Mike Zabrocki can be used to express the image of a Schur function under the Bergeron-Garsia Nabla operator as a weighted sum of a suitable collection of ``Parking Functions.'' The validity of these expressions is, of course, going to be conjectural until the compositional refinement of the Shuffle Conjecture is established.

Keywords

• parking function
• nabla
• hall-littlewood operators
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]