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Number of partitions of 2N into N parts

  1. Aug 17, 2010 #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.
     
  2. jcsd
  3. Aug 17, 2010 #2
    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.
     
  4. Aug 17, 2010 #3
    That takes care of it. Thanks!
     
  5. Aug 17, 2010 #4
    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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