Number of partitions of 2N into N parts

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Discussion Overview

The discussion revolves around the number of partitions of an even number 2N into N parts and its relationship to the number of partitions of N. Participants explore whether this relationship is known and seek references or proofs related to this concept.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests that the number of partitions of 2N into N parts is equal to the number of partitions of N and asks for references to support this claim.
  • Another participant provides a reasoning based on a bijective proof using Ferrer's diagrams, stating that the number of partitions of any integer m into j parts equals the number of partitions of m with the largest part j, leading to the conclusion that the number of partitions of 2n into n parts equals the number of partitions of 2n with largest part n.
  • A later reply expresses satisfaction with the explanation provided, indicating that it resolves the initial inquiry.
  • Another participant reiterates the bijective proof and extends the discussion to a more general case, proposing that for positive integers m and j, with m > j >= m/2, the number of partitions of m into j parts is equal to the number of partitions of m-j.

Areas of Agreement / Disagreement

While there is a general agreement on the validity of the bijective proof presented, the initial claim about the relationship between the partitions of 2N and N remains open for further validation as no explicit consensus on its established status is reached.

Contextual Notes

The discussion does not clarify any assumptions or limitations regarding the definitions of partitions or the conditions under which the bijective proofs hold.

FaustoMorales
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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.
 
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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!
 
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.

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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