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Given a positive integer \$n\$, a partition of \$n\$ is a ascending sequence of numbers that sum to \$n\$.
Given two partitions \$a\$ and \$b\$, \$a\$ is a refinement of \$b\$ iff \$b\$ can be created by combining elements of \$a\$. (The elements do not need to be adjacent.)
For example 1+2+3+4+7...