The University of Michigan Combinatorics Seminar
Level-restricted paths play an important role in the study of exactly solvable models in statistical mechanics. In the context of crystal base theory they correspond to certain highest weight vectors of modules of quantum affine algebras. Using a sign-reversing involution we obtain a formula for the $q$-enumeration of these paths. For type $A$ the resulting polynomial is the (generalized) Kostka polynomial, and our results prove a conjecture by Foda, Leclerc, Okado, and Thibon. This is work in collaboration with Mark Shimozono.