Face-Counting Identities and Derivatives of Face Polynomials

In this paper, we first introduce two incidence matrices related to the vertex-deck and the edge-deck of a given simplicial complex. Then, we obtain two face-counting identities based on these incidence matrices. Using these face-counting identities, we obtain combinatorial interpretations of the first and the second derivatives of face polynomials of simplicial complexes. Finally, we propose several interesting open questions and conjectures.