Math 2.0

Number theory Theorem

Generating function for partitions

Builds on Partition functionGenerating function

Statement

Let p(n)p(n) be the partition function. Then, as formal power series in ℤ[ ⁣[x] ⁣]\mathbb{Z}[\![x]\!], ∑n≥0p(n) xn=∏k≥111−xk.\sum_{n \ge 0} p(n)\, x^n = \prod_{k \ge 1} \frac{1}{1 - x^k}.

Remarks

The infinite product makes sense, because each coefficient of the product depends only on finitely many factors. Proof idea: the factor 11−xk=1+xk+x2k+…\frac{1}{1 - x^k} = 1 + x^k + x^{2k} + \dots chooses how many parts equal kk. Restricting the factors gives other results. For example, ∏k≥1(1+xk)\prod_{k \ge 1} (1 + x^k) counts partitions into distinct parts. This gives Euler's partition theorem.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Formal power series, definition, no evidence levelFormal power seriesdefinitionInteger partition, definition, no evidence levelInteger partitiondefinitionGenerating function, definition, no evidence levelGenerating functiondefinitionPartition function, definition, no evidence levelPartition functiondefinitionGenerating function for partitions, statement, no evidence levelGenerating function f…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Formal power series definition
  2. Integer partition definition
  3. Generating function definition
  4. Partition function definition
  5. Generating function for partitions statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Partition_function_(number_theory), Generating function
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