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This paper extends to finite cell complexes the classical enumerative combinatorics of graphs and their forests in generalized Kirchoff type theorems. Four types of combinatorial formulae are obtained involving summations over the resepctive notions introduced here for cell complexes which are forests, coforests, and two dual notions. Special cases of these formulae include counting spanning trees in cell complexes, for which some of the formulae are due to M. Catanzaro, V. Chernyak, and J. Klein. Combining together a relation of theirs to Reidemeister-Franz torsion with a second relation to this torsion introduced here yields a generalization to cell complexes of Trent's classical theorem on mesh matrices for graphs.

Enumerative Combinatorics of Simplicial and Cell Complexes: Kirchhoff and Trent Type Theorems