Is there some sort of explicit (ideally polynomial) characteristic function of simplicial complexes?
Let $C = \{\{1, 2, 3\}, \{1,2\}, \{2, 3\}, \{1,3\}, \{3, 4\}, \{1\}, \{2\}, \{3\}, \{4\}, \emptyset\}$. Here we have a filled triangle $(1,2,3)$ and an edge $(3,4)$. I'm looking for indicator function $f(s) = 1 \Leftrightarrow s \in C$