Show that for any positive integer $n$, the number of partitions of $n$ in which the two largest parts are equal is $p(n) − p(n − 1)$.
What I have so far:
We can construct a bijection from the set of all partitions of $n-1$ onto the set of all partitions of $n$ that have at least one part equal to one. The bijection is just simply adding a part equal to $1$ to the end of each partition of $n-1$.
What else?
Best Answer
HINT: Notice that the result is the same as saying that there are $p(n-1)$ partitions of $n$ whose two largest parts are different, so let’s try to prove that instead. Let $p_1+p_2+\ldots+p_m$ be such a partition, where as usual $p_1\ge p_2\ge\ldots\ge p_m$; then
$$(p_1-1)+p_2+\ldots+p_m$$
is a partition of $n-1$. Is this correspondence reversible?