Found this question on a list of past USAMO problems- and the one of the few that I couldn't solve. At a quick glance, seems like a deep problem.
P(n) is to partition an integer n greater than 1 so that n can be represented by the addition of positive integers in non decreasing order: P(4): 1+1+1+1, 1+1+2, 1+3, 2+2 and 4).
For any partition P(n), let A(n) denote NUMBER of 1's in each partition. For any partition P(n), let B(n) equal the NUMBER of distinct integers in each partition.
Why is the sum of A(n) = the sum of B(n)?
(i.e) P(4): 1+1+1+1, 1+1+2, 1+3, 2+2 and 4 Sum of A(n)= 4+ 2+1+0+0= 7 Sum of B(n)=1+2+2+1+1=7 I need a combinatorial interpretation of the solution.
