# Enumeration of a certain sequence

Denote by $$a(n)$$ the number of families $$\mathcal{F} \subseteq \mathcal{P}(X)$$ on a finite set $$X$$ with $$n$$ elements satisfying:

• $$\emptyset, X \in \mathcal{F}$$
• For all $$U, V \in \mathcal{F}$$ it holds true that $$U \cap V \in \mathcal{F}$$.
• For all nonempty disjoint $$U, V \in \mathcal{F}$$ there exists $$H \in \mathcal{F}$$ such that it holds true that $$X \setminus H \in \mathcal{F}$$ and $$U \subseteq H$$ and $$V \subseteq X \setminus H$$.

If we were to drop the last restriction, the sequence is given in OEIS as A102894.

Relevant question.

I am a mathematician with no serious programming capabilities. Can someone help me and write a code to enumerate the sequence $$a(n)$$?

• Isn't this the same question as mathematica.stackexchange.com/questions/271425/…? Aug 5 at 18:25
• No, the third condition is different and I can’t quite relate the code to this problem. Aug 5 at 18:26
• Unless I'm missing something, isn't the last condition impossible to satisfy? Let $U,V=X\in\mathcal{F}$. Then $V=X\subseteq X\setminus H$ implies $H=\varnothing$, but $U=X\not\subseteq\varnothing$? yesterday
• You are right. I corrected my question. yesterday