We say that $G \in P(P(S))$ is a "game" on $S$ if for any $A,B \subseteq S$, we have $A \in G \iff \overline A \notin G$ and $A \subseteq B \implies (A \in G \implies B \in G)$.
Examples:
- The moves in a game of hex, or variants.
- Any non-trivial filter
- Upperwards-closed sets on a boolean algebra
(Note: To play the game, you place cards on a table with elements of $S$. Players take turn taking cards, and if you collect a set of cards in $G$, you win.)
Games $G_1, G_2$ are considered the same if there is some bijection $i : S_1 \to S_2$ such that ${i^\to}^\to(G_1) = G_2$ ($f^\to$ means the image of $f$, btw).
My question is, if $|S| = k \in \mathbb N$, how many distinct games are there on $S$?