Let $\mathcal{F}\subseteq2^{[n]}$ be a union-closed family of sets. For a set $S\in[n]$ (not necessary belong to $\mathcal{F}$), define $w_{\mathcal{F}}(S)$ to be the number of subset of $S$ which belong to $\mathcal{F}$. For $A\in\mathcal{F}$, consider the quantity: $$q(A)=\sum_{S\subseteq A}(-1)^{|A|-|S|}w_{\mathcal{F}}(S)\log_2w_{\mathcal{F}}(S)$$.
Does there exist a combinatorial/probabilistic interpretation of $q(A)$ or $2^{q(A)}$? Note that we have a simple interpretation for $p_k(A)=\sum_{S\subseteq A}(-1)^{|A|-|S|}w^k_{\mathcal{F}}(S)$, which is the number of $k$-tuple $A_1,A_2,...,A_k\in\mathcal{F}$ such that $\bigcup_{i=1}^kA_i=A$.
