Two Kinds Of Partitions

In general, let $A_{n}$ be the set of partitions of $n$ and $B_{n}$ be the number of partitions of $2n$ into $n$ parts.

Now form the set $A_{n}'$ , which has the same number of elements as $A_{n}$ but for each element $a$ of $A_{n}$ with fewer than $n$ parts we form a corresponding element $a'$ of $A_{n}'$ by adding sufficient $0$ 's at the end of $a$ so that $a'$ has $n$ parts. For example, if $2 + 2 + 1$ is an element of $A_{5}$ then $2 + 2 + 1 + 0 + 0$ would be the corresponding element of $A_{5}'$ . (Any element of $A_{n}$ with $n$ parts remains unchanged when "transferred" to $A_{n}'$ .)

Now for each element $a'$ of $A_{n}'$ we can add $1$ to each of the $n$ parts of $a'$ to create a partition of $2n$ into $n$ parts, i.e., an element $b$ of $B_{n}$ . Also, for each element $b$ of $B_{n}$ we can subtract $1$ from each of the parts of $b$ to create an element of $A_{n}'$ . So we have a one-to-one correspondence between the elements of $A_{n}'$ and $B_{n}$ , which implies that the two sets have the same number of elements.

But since $A_{n}$ and $A_{n}'$ also have the same number of elements, we see that $p(n) = q(n)$ , and thus $p(12) - q(12) = \boxed{0}$ .

It is a known result that the number of partitions of n into exactly k parts is equal to the number of partitions of n where k is the largest part, and this can be proved by a bijection from the Ferrer's diagram. So q(n)=p(n) for all n according to definition of p and q.