  We introduce the notion of P-polynomial coherent configurations and show that
they can have at most two fibres. We then introduce a class of two-fibre
coherent configurations which have two distinguished bases for the coherent
algebra, similar to the Bose-Mesner algebra of an association scheme. Examples
of these bipartite coherent configurations include the P-polynomial class of
distance-biregular graphs, as well as quasi-symmetric designs and strongly
regular designs.
