Rational Parking Functions and LLT Polynomials

TL;DR

This paper proves that the combinatorial interpretation of the Rational Shuffle Conjecture yields a Schur-positive symmetric polynomial and relates rational Dyck paths to skew LLT polynomials, generalizing previous results.

Contribution

It establishes a new connection between rational Dyck paths and skew LLT polynomials, extending prior work on the Shuffle Conjecture.

Findings

The combinatorial side of the Rational Shuffle Conjecture is Schur-positive.

Rational Dyck paths' contributions can be computed as skew LLT polynomials.

Explicit description of the skew diagram in terms of (m,n)-cores.

Abstract

We prove that the combinatorial side of the "Rational Shuffle Conjecture" provides a Schur-positive symmetric polynomial. Furthermore, we prove that the contribution of a given rational Dyck path can be computed as a certain skew LLT polynomial, thus generalizing the result of Haglund, Haiman, Loehr, Remmel and Ulyanov. The corresponding skew diagram is described explicitly in terms of a certain (m,n)-core.