Paper detail

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

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 $<\nabla e_n, h_μ>$ where \nabla is the Macdonald polynomial eigen-operator of Bergeron and Garsia and $h_μ$ is the homogeneous basis indexed by $μ=(μ_1,μ_2,...,μ_k)$ 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 $μ_1,μ_2,...,μ_k$. 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.