  Using the expansion of the inverse of the Kostka matrix in terms of tabloids
as presented by Egecioglu and Remmel, we show that the fusion coefficients can
be expressed as an alternating sum over cylindric tableaux. Cylindric tableaux
are skew tableaux with a certain cyclic symmetry. When the skew shape of the
tableau has a cutting point, meaning that the cylindric skew shape is not
connected, or if its weight has at most two parts, we give a positive
combinatorial formula for the fusion coefficients. The proof uses a slight
modification of a sign-reversing involution introduced by Remmel and Shimozono.
We discuss how this approach may work in general.
