比利时vs摩洛哥足彩
,
university of california san diego
****************************
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
****************************