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Department of Mathematics,
University of California San Diego

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Math 269 - Combinatorics

Emily Leven

UCSD

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

Abstract:

The original Shuffle Conjecture of Haglund et al. [2005] has a symmetric function side and a combinatorial side. The symmetric function side may be simply expressed as $\big\langle \nabla e_n \, , \, h_{\mu} \big\rangle$ where $\nabla$ is the \hbox{Macdonald} polynomial eigen-operator of Bergeron and Garsia [1999] 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 $1 2 3 \cdots n$ 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.

Jeff Remmel

April 9, 2013

4:00 PM

AP&M 7321

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