Two Kinds Of Partitions

Probability Level 3

Let $p(n)$ be the number of partitions of $n$ . Let $q(n)$ be the number of partitions of $2n$ into exactly $n$ parts. For example, $q(3) = 3$ because $6 = 4+1+1 = 3+2+1 = 2+2+2.$ Compute $p(12)-q(12).$

Definition: A partition of an integer is an expression of the integer as a sum of one or more positive integers, called parts. Two expressions consisting of the same parts written in a different order are considered the same partition ("order does not matter").