The number of partitions of an even integer 2N into N parts is definitively equal to the number of partitions of N. This conclusion is supported by a bijective proof utilizing Ferrer's diagrams, which demonstrates that the number of partitions of 2N with the largest part N corresponds directly to the partitions of N. Additionally, a more general principle is established: for positive integers m and j, where m > j >= m/2, the number of partitions of m into j parts equals the number of partitions of m-j.
PREREQUISITESMathematicians, combinatorial theorists, and students studying number theory or partition theory will benefit from this discussion.
JCVD said:The number of partitions of any integer m into j parts equals the number of partitions of m with largest part j (easy bijective proof using Ferrer's diagrams). Thus the number of partitions of 2n into n parts equals the number of partitions of 2n with largest part n. The bijection between partitions of 2n with largest part n and partitions of n is straightforward.