Our professor gave us this task: let $T_0^{(n)},T_1^{(n)},S^{(n)},L^{(n)},M^{(n)}$ be the sets of boolean functions $\{0,1\}^n\rightarrow\{0,1\}$ which preserve $0$, preserve $1$, are self-dual, are linear and are monotonous, respectively. Find $|T_0^{(n)}|,|T_1^{(n)}|,|S^{(n)}|,|L^{(n)}|,|M^{(n)}|.$
It was easy to calculate $|T_0^{(n)}|=2^{{2^n}-1}=|T_1^{(n)}|$ and $|L^{(n)}|=2^{n+1}$ using algebraic normal forms. $|S^{(n)}|=2^{2^{n-1}}$ was not difficult to compute.
But what about $|M^{(n)}|$? I see that we shall find the number of free distributive lattices on $n$ generators but do not know, how to do this. I've counted that for $n=0,1,2,$ and $3$ $|M^{(n)}|$ equals $2,3,6$ and $20$, respectively.
Any help is appreciated! :-)
The sequence you're talking about is called 'Dedekind numbers'. Here's an article about them: https://en.wikipedia.org/wiki/Dedekind_number. Even though the problem of their computation is almost 125 years old, an explicite formula is still not found.
https://www.sfu.ca/~tstephen/Papers/r7.pdf