  We introduce, in a systematic way, a set of generating functions that solve
all the different combinatorial problems that crop up in the study of black
hole entropy in Loop Quantum Gravity. Specifically we give generating functions
for: The different sources of degeneracy related to the spectrum of the area
operator, the solutions to the projection constraint, and the black hole
degeneracy spectrum. Our methods are capable of handling the different
countings proposed and discussed in the literature. The generating functions
presented here provide the appropriate starting point to extend the results
already obtained for microscopic black holes to the macroscopic regime --in
particular those concerning the area law and the appearance of an effectively
equidistant area spectrum.
