Number of partitions of 2N into N parts

  • #1
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

  • #2
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0
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.
 
  • #3
That takes care of it. Thanks!
 
  • #4
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.
 

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