Face-Counting Identities and Derivatives of Face Polynomials

Year 2023, , 443 - 448, 31.12.2023

Hossein Teimoori Faal

https://doi.org/10.47000/tjmcs.1120399

Abstract

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 present combinatorial interpretations of the first and the second derivatives of face polynomials of simplicial complexes. We also propose several interesting open questions and conjectures. Finally, we conclude the paper with a discussion about some possible future research works.

Keywords

Face polynomial, face-counting identities, incidence matrices, derivatives

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