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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 $a(n)$?

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    $\begingroup$ Isn't this the same question as mathematica.stackexchange.com/questions/271425/…? $\endgroup$ Aug 5 at 18:25
  • $\begingroup$ No, the third condition is different and I can’t quite relate the code to this problem. $\endgroup$ Aug 5 at 18:26
  • $\begingroup$ 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 $U=X\not\subseteq\varnothing$? $\endgroup$
    – Lukas Lang
    yesterday
  • $\begingroup$ You are right. I corrected my question. $\endgroup$ yesterday

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