# Number of partitions of 2N into N parts

The number of partitions of an even number 2N into N parts appears to be equal to the number of partitions of N.

Is this known? If so: Can anyone provide a reference of the corresponding proof? Thanks in advance for any information on this.

## Answers and Replies

Related Linear and Abstract Algebra News on Phys.org
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.

That takes care of it. Thanks!

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.
This clever idea can be used to prove something more general: Given positive integers m and j, with m > j >= m/2, the number of partitions of m into j parts is the number of partitions of m-j.