Count of Commutative Binary Operations with Identity

Theorem
Let $S$ be a set whose cardinality is $n$.

The number $N$ of possible different commutative binary operations that can be applied to $S$ which have an identity element is given by:


 * $N = n^{\frac {n \paren {n - 1} } 2 + 1}$

Proof
From Count of Commutative Binary Operations with Fixed Identity, there are $n^{\frac {n \paren {n - 1} } 2}$ such binary operations for each individual element of $S$.

As Identity is Unique, if $x$ is the identity, no other element can also be an identity.

As there are $n$ different ways of choosing such an identity, there are $n \times n^{\frac {n \paren {n - 1} } 2}$ different algebraic structures with an identity.

These are guaranteed not to overlap by the uniqueness of the identity.

Hence the result.