What are the feasible $2^n$-tuples of entropies $h(S) := H(X_{i_1},\dots,X_{i_{|S|}})$ where $X_1,\dots,X_n$ are discrete random variables with some (unknown) joint probability distribution as $S=\{i_1,\dots,i_{|S|}\}$ ranges over the subsets of $\{1,\dots,n\}$? Here $X_{i_1},\dots,X_{i_{|S|}}$ denotes the compound random variable that in classic information theory texts might be written as $XY$, $XYZ$, etc. and $H(\cdot)$ is the standard entropy $\sum p \log 1/p$.

Of course $h(\{\,\}) = 0$ and $S_1 \subseteq S_2$ implies $h(S_1) \leq h(S_2)$; furthermore, nonnegativity of conditional mutual information gives $$h(S_1 \cup S_2 \cup S_3) + h(S_3) \leq h(S_1 \cup S_3) + h(S_2 \cup S_3).$$ Are there other constraints, linear or otherwise?

Assuming the answer is “no”, I’d be interested in knowing a combinatorial characterization of the extreme rays of the convex polyhedral cone of achievable $2^n$-tuples.