A Refinement Of The Shuffle Conjecture With Cars Of Two Sizes And T = 1/Q

The original Shuffle Conjecture of [12] has a symmetric function side and a combinatorial side. The symmetric function side may be simply expressed as where del is the Macdonald polynomial eigen-operator of [3] and h(mu) is the homogeneous basis indexed by mu = (mu(1), mu(2), ..., mu(k)) proves n. The combinatorial side q, t-enumerates a family of Parking Functions whose reading word is a shuffle of k successive segments of 123 ... 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.