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 following: 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.

• Received 30 April 2008

DOI:https://doi.org/10.1103/PhysRevD.77.121502