Count of Commutative Operations on Finite Set which are Closed on Every Subset

Theorem
Let $S$ be a finite set with $n$ elements.

There are $2^{n \paren {n - 1} / 2}$ commutative binary operations on $S$ which are closed on all subsets of $S$.

Proof
From Count of Operations on Finite Set which are Closed on Every Subset, there are $2^{n \paren {n - 1} }$ binary operations on $S$ which are closed on all subsets of $S$.

For each $a, b \in S$, we have that
 * there are $2$ possible products for $a \circ b$
 * there are $2$ possible products for $b \circ a$

and so on the set $\set {a, b}$ there are $4$ binary operations which are closed on all subsets of $\set {a, b}$.

Let $\circ$ be commutative.

Then:
 * $a \circ b = b \circ a$

and so now there are only $2$ possible such products.

Hence the number of commutative binary operations on $S$ which are closed on all subsets of $S$ equals the number of distinct doubleton subsets of $S$.

There are half as many of these as there are ordered pairs.

That is, there are $\dfrac {n^2 - n} 2 = \dfrac {n \paren {n - 1} } 2$ doubletons $\set {x, y} \subseteq S$, such that $x \ne y$.

The result follows.