Set of unordered integer partitions

Definition
Let $$n$$ be a nonnegative integer. An unordered integer partition of $$n$$ is an additive partition of $$n$$ into positive integers, without any specific ordering on the parts. The set of unordered integer partitions of $$n$$, sometimes denoted $$P(n)$$, is the set of all such unordered integer partitions.

The cardinality of this set, also termed the number of unordered integer partitions or partition number, is denoted $$p(n)$$. We have:

$$p(n) = O(e^{\pi\sqrt{2n/3}})$$.

The set of unordered integer partitions figures in the following ways:


 * It is in canonical bijection with the set of conjugacy classes in the symmetric group of degree $$n$$. The bijection is via the cycle type map, and it is a bijection because cycle type determines conjugacy class.
 * It is in canonical bijection with the set of irreducible representations over the rationals (and also, over any algebraic extension of the rationals) of the symmetric group of degree $$n$$.
 * For any fixed prime number $$p$$, it is in canonical bijection with the set of isomorphism classes of abelian groups of order $$p^n$$, via the structure theorem for finitely generated abelian groups.

Examples
We have the following small values: