  Replacing the usual notion of quotient sets by the notion of orbiquotient
sets we obtain a generalization of P\'olya theory. The key ingredient of our
extended theory is the definition of the orbicycle index polynomial which we
compute in several examples. We apply our theory to the study of orbicycles on
orbiquotient sets.
  Keywords: Orbifolds, P\'olya Theory, Partition Lattice.
