We now have a partition of 10 with 3 parts. Now we can see that if we start off with a diagram of
a partition whose largest part is r and count by the columns instead of by the rows we will end up
with a partition of n with exactly r parts.
This now suggests a way to show a one-to-one correspondence between the two sets. Take a
partition of n into exactly r parts, i.e. p1 + p2 + · · · + pr. Now make the partition q1 + q2 + · · · + qk
where each qi represents the number of (pj )’s as large as i. For example, if we started with 5+3+2+2
we would end up with 4 + 4 + 2 + 1 + 1. But now notice that we start off with a partition into
exactly r parts meaning that the new partition will have addends which are at most r. Thus every
partition of n into exactly r parts directly corresponds to one partition of n whose largest addend
is r.