@HDB: Regarding your comment of 2023-04-03 13:13:59Z, it seems to me that an abstract simplicial complex on $[n] := \{0,\ldots,n\}$ is precisely the same thing as a simplicial subset of $\Delta^n$ (considering the nondegenerate simplices, in both cases the condition is that if you take a $k$-simplex you must take all its $\ell$-simplices for $\ell\leq k$). So your formula for $\Omega(n)$ is correct, and proves what is stated in this answer.
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