it is known by Weyl character formula that multiplicity of the weight $\wedge_{0} n.\delta$ in the basic representation $L(\wedge_{0})$ of $\hat{sl_2} $is equal to the number of partitions of $n$.It is known that LS(Lakshmibai seshadri) paths of shape $\wedge_{0}$ gives a parametrization for the basis of $L(\wedge_{0})$.There is a paper of Yasmine B.Sanderson on dimensions of Demazure modules for rank two affine algebras where she has found LS paths for the basic representation of $\hat{sl_2}$ explicitly. Here is the explicit description of paths. The LS Paths of shape $\wedge_{0}$ are those paths $\pi = (\sigma,a) $ such that $\sigma:w_{n+k} > w_{n+k1} > w_{n+k2} >...> w_{n} $ where $w_i$ is the product of $i$ reflections $r_{0}.r_{1}.r_{0}.r_{1}...$ and $ n,k \geq 0 $ and $a:0 < a_{n+k} < a_{n+k1} < ...... < a_{n+1} < 1$ where $ a_j.d_j\epsilon Z $ and $d_j = j$ and $a_{i} \epsilon Q$ Now the question is how can one show that there are exactly $P(n)$ of LS paths of shape $\wedge_{0}$ whose weight is $\wedge_{0}  n.\delta$ ?
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