Math 2.0

Combinatorics Theorem

Euler's partition theorem

Builds on Integer partition

Statement

Let nn be a natural number. Then the number of partitions of nn into odd parts equals the number of partitions of nn into distinct parts.

Remarks

For n=5n = 5, the partitions into odd parts are 55, 3+1+13 + 1 + 1 and 1+1+1+1+11 + 1 + 1 + 1 + 1. The partitions into distinct parts are 55, 4+14 + 1 and 3+23 + 2. A proof with generating functions: ∏k≥1(1+xk)=∏k≥11−x2k1−xk=∏k odd11−xk\prod_{k \ge 1} (1 + x^k) = \prod_{k \ge 1} \frac{1 - x^{2k}}{1 - x^k} = \prod_{k \text{ odd}} \frac{1}{1 - x^k}. Glaisher gave a bijection. It merges equal parts in pairs, again and again, until all parts are distinct.

Reading path

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Further reading

  1. Web https://en.wikipedia.org/wiki/Integer_partition, Odd parts and distinct parts
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