The component counts of random injections

A model of random injections is defined which has domain A boolean OR B and codomain A boolean OR C, where A, B and C are mutually disjoint finite sets such that vertical bar B vertical bar <= vertical bar C vertical bar. The model encompasses both random permutations, which is the case B = C = empty set, and random maximum matchings of a complete bipartite graph, which is the case A = empty set. The possible components of random injections are cycles and paths. Results on the counts of cycles and paths of different sizes are obtained for this model.