Math 2.0

Number theory Definition

Partition function

Builds on Integer partition

Definition

The partition function is the function p:ℕ→ℕp : \mathbb{N} \to \mathbb{N} such that p(n)p(n) is the number of partitions of nn.

Remarks

The values for n=0,1,…,10n = 0, 1, \dots, 10 are 1,1,2,3,5,7,11,15,22,30,421, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42. The generating function for partitions is ∑n≥0p(n)xn=∏k≥111−xk\sum_{n \ge 0} p(n) x^n = \prod_{k \ge 1} \frac{1}{1 - x^k}. The function grows faster than any polynomial and slower than any exponential cnc^n with c>1c > 1.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph
Integer partition, definition, no evidence levelInteger partitiondefinitionPartition function, definition, no evidence levelPartition functiondefinition
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.
  • no evidence level
  1. Integer partition definition
  2. Partition function definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Partition_function_(number_theory), Definition and examples
About this page
Tag001CX45SWQ
URIm2://001CX45SWQ
Kinddefinition
WritingW1 stated
MSC11P81
DefinesPartition function
UsesInteger partition

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by