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 $ \langle\nabla e_n \, , \, h_{\mu}\rangle$ where $\nabla$ is the Macdonald polynomial eigen-operator of [3] and $h_\mu$ is the homogeneous basis indexed by $\mu=(\mu_1,\mu_2,\ldots,\mu_k) \vdash n$. The combinatorial side $q,t$-enumerates a family of Parking Functions whose reading word is a shuffle of $k$ successive segments of $123\ldots n$ of respective lengths $\mu_1,\mu_2,\ldots,\mu_k$. 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.

Keywords

parking functions, shuffle conjecture, dinv, Narayana numbers

2010 Mathematics Subject Classification

Primary 05E05. Secondary 05E10.

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Published 12 February 2014