We present generalized versions of the concepts of seniority number and ionicity. These generalized numbers count respectively the partially occupied and fully occupied shells for any partition of the orbital space into shells. The Hermitian operators whose eigenspaces correspond to wave functions of definite generalized seniority or ionicity values are introduced. The generalized seniority numbers afford to establish refined hierarchies of configuration interaction spaces within those of fixed ordinary seniority. Such a hierarchy is illustrated on the buckminsterfullerene C 60 molecule.