A refinement of the Shuffle Conjecture with cars of two sizes and $t=1/q$

TL;DR

This paper proves a combinatorial refinement of the Shuffle Conjecture for two cars of different sizes at t=1/q, resulting in a q-analogue of Narayana numbers, connecting symmetric functions and parking functions.

Contribution

It provides a proof of a refined version of the Shuffle Conjecture for the case k=2 and t=1/q, linking symmetric functions with combinatorial parking functions.

Findings

Reduction to q-binomial coefficients at t=1/q

Proof of the refinement for k=2

Derivation of a q-analogue of Narayana numbers

Abstract

The original Shuffle Conjecture of Haglund et al. has a symmetric function side and a combinatorial side. The symmetric function side may be simply expressed as $<\nabla e_n, h_{\mu}>$ where \nabla is the Macdonald polynomial eigen-operator of Bergeron and Garsia and $h_\mu$ is the homogeneous basis indexed by $\mu=(\mu_1,\mu_2,...,\mu_k)$ partitions of n. The combinatorial side q,t-enumerates a family of Parking Functions whose reading word is a shuffle of k successive segments of 1,2,3,...,n of respective lengths $\mu_1,\mu_2,...,\mu_k$. It can be shown that for t=1/q the symmetric function side reduces to a product of q-binomial coefficients and powers of q. This reduction suggests a surprising combinatorial refinement of the general Shuffle Conjecture. Here we prove this refinement for k=2 and t=1/q. The resulting formula gives a q-analogue of the well studied Narayana numbers.