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abstract: '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 $\mathfrak{sl}_2[t]$. 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 coeffficients of Ramanujan’s fifth order mock theta functions $\phi_0, \phi_1, \psi_0,  \psi_1$, Kostka polynomials for hook partitions and quotients of Chebyshev polynomials. We also get 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 $B_n \text{ and } G_2$.'
address:
- 'Université Laval, Départment de mathématiques et de Statistique, Québec, QC, Canada'
- 'University of Bochum, Faculty of Mathematics, Universit[ä]{}tsstr. 150, 44801 Bochum, Germany'
author:
- Rekha Biswal
- Deniz Kus
bibliography:
- 'bibfile.bib'
title: A combinatorial formula for graded multiplicities in excellent filtrations
---

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Introduction
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In this paper, we are interested in objects in the category of finite-dimensional ${\mathbb Z}$–graded 