Consider $n$-dimensional vector space $\mathbb{F}_p^n$ over finite field $\mathbb{F}_p$ and a function $a: \mathbb{F}_p^n\rightarrow\mathbb{C}$. Let $$ F(a) = \sum_{\substack{S\subset \mathbb{F}_p^n \\ S \ \text{is linearly independent}}} \prod_{v \in S}a(v),$$

$$ G(a) = \sum_{\substack{S\subset \mathbb{F}_p^n \\ S \ \text{is linearly independent} \\ |S|=n}} \prod_{v \in S}a(v).$$

For example, if $M$ is multiset of vectors and $a(v)$ is number of vectors $v$ in $M$, $F(M)$ and $G(M)$ are number of linearly independent subsets and bases in $M$, respectively.

Do $F$ and $G$ have any nice combinatorical and/or algebraic properties (factorization, effective methods of calculation, connection to exterior algebra etc.)? Do they connected to combinatorical lattice theory somehow?


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