Dedekind number

Remarks

Identify $\set{0, 1}^n$ with the subsets of $\set{1, \ldots, n}$. A monotone $f$ is determined by the minimal sets $S$ with $f(S) = 1$, and these form an antichain. So $M(n)$ is also the number of antichains of subsets of $\set{1, \ldots, n}$, including the empty antichain.

Example: $M(2) = 6$. The monotone functions of $x, y$ are $0$, $1$, $x$, $y$, $x \wedge y$ and $x \vee y$. The matching antichains are $\emptyset$, $\set{\emptyset}$, $\set{\set{1}}$, $\set{\set{2}}$, $\set{\set{1, 2}}$ and $\set{\set{1}, \set{2}}$.

Non-example: the function $x \oplus y$, equal to $1$ exactly when $x \ne y$, is not monotone. Indeed $(1, 0) \le (1, 1)$ but $f(1, 0) = 1 > 0 = f(1, 1)$.