Set of self-conjugate unordered integer partitions

Definition
Let $$n$$ be a nonnegative integer. A self-conjugate unordered integer partition of $$n$$ is an unordered integer partition of $$n$$ whose Ferrers diagram (or Young diagram) is self-conjugate. In other words, it is an unordered integer partition that is equal to its conjugate partition.

The set of self-conjugate unordered integer partitions is in canonical bijection with the following sets: