Intersection matrices and the Johnson scheme

In this paper we introduce a generating function F t sk(v)(z) which is a polynomial in terms of z whose coefficients are some intersection matrices. This provides a uniform framework in which several intersection matrices can be extracted from F t sk(v)(z). Several nice properties of F t sk(v)(z) are derived by applying the operator d/dz and studying the operator zd/dz. In the new framework some well-known identities on intersection matrices arise as natural consequences. As an application two new bases for the Johnson scheme are constructed and the eigenvalues of a family of intersection matrices which contains the adjacency matrices of the Johnson scheme are derived. Finally, we determine the rank of some intersection matrices. AMS Classification: 05B20; 05E30