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

摘要

The original Shuffle Conjecture of [12] has a symmetric function side and a combinatorial side. The symmetric function side may be simply expressed as <▽e_n, h_μ> where ▽ is the Macdonald polynomial eigen-operator of [3] and h_μ is the homogeneous basis indexed by μ = (μ_1, μ_2,..., μ_k) ┝ 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 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 = l/q. The resulting formula gives a q-analogue of the well-studied Narayana numbers.