Two Kinds Of Partitions
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").