Tensor products and dimensions of simple modules for symmetric groups

LetK be an algebraically closed field of characteristic,p>0 and letD ^λ be the simple modules of the symmetric groupS _r overK where λ is a p-regular partition ofr. The dimensions ofD ^λ for λ with at mostn parts are the same as the multiplicities of direct summands ofD ^⊗r whereE is the natural module for the groupGL _n (K). Whenn=2 we determine generating functions for these multiplicities and hence for the dimensions ofD ^λ for all partitions λ with two parts. These can be expressed as rational functions of Chebyshev polynomials; and we obtain explicit formulae for the coefficients.