For integers n, k > 0 let
• P1 be the number of partitions of n.
• P2 be the number of partitions of In + k, where n + k is the greatest summand.
• P3 be the number of partitions of 2n + k into precisely n + k summands.
Using the concept of the Ferrers graph, prove that P1 = P2 and P2 = P3, thus concluding that the number of partitions of 2n + k into precisely n + k summands is the same for all k.