A filtration of a representation whose successive quotients are isomorphic to Demazure modules is called an excellent filtration. In this paper we study graded multiplicities in excellent filtrations of fusion products for the current algebra \( {\mathfrak{sl}}_2\left[t\right] \). We give a combinatorial formula for the polynomials encoding these multiplicities in terms of two-dimensional lattice paths. Corollaries to our main theorem include a combinatorial interpretation of various objects such as the coefficients of Ramanujan’s fifth order mock theta functions ϕ₀, ϕ₁, ψ₀, ψ₁, Kostka–Foulkes polynomials for hook partitions and quotients of Chebyshev polynomials. Moreover, a combinatorial interpretation of the graded multiplicities in a level one flag of a local Weyl module associated to the simple Lie algebras of type B_n and G₂ is obtained.

其连续商数与Demazure模块同构的表示形式的过滤称为出色过滤。在本文中，我们研究了当前代数\（{\ mathfrak {sl}} _ 2 \ left [t \ right] \）在融合产品的出色过滤中的分级多重性。我们根据二维晶格路径为编码这些多重性的多项式给出了一个组合公式。推论我们的主要定理包括各种物体如拉马努金的五阶模拟theta函数的系数的组合演绎φ ₀，φ ₁，ψ ₀，ψ ₁，Kostka–Foulkes多项式用于切比雪夫多项式的钩子分区和商。此外，获得了与类型B _n和G ₂的简单李代数相关联的局部Weyl模的一级标志中的多重多重性的组合解释。