Mathematics / Statistics · BEGINNER LESSON
Permutation
A permutation is an ordered rearrangement of the elements of a set, or equivalently a function from a non-empty set to itself that is one-to-one and onto. For the set 1, 2, ..., n, it can be written by showing where each position is sent.
What a permutation is
A permutation is a rearrangement of the elements of a set in which each element is used once and only once. In function language, it is a one-to-one and onto mapping from a non-empty set to itself, which means every element is matched to exactly one element in the same set, and every target is hit. For the integers 1 through n, the rearrangement can be written as the values of the function at 1, 2, ..., n.
[1] dlmf.nist.gov · 26DLMF: §26.2 Basic Definitions ‣ Properties ‣ Chapter 26 ... [3] Permutation - Glossary | CSRC
How it works
For the set {1, 2, 3}, the permutation written as 231 means 1 is sent to 2, 2 is sent to 3, and 3 is sent to 1. This records the new position of each item. A cycle is one way to describe part of a permutation: for example, if 1 goes to 2, 2 goes to 5, and 5 goes back to 1, then those three elements form a cycle, while other elements may be fixed or swapped separately.
[1] dlmf.nist.gov · 26DLMF: §26.2 Basic Definitions ‣ Properties ‣ Chapter 26 ...
Statistical use and limits
In statistics, a permutation test builds a reference distribution by rearranging data in ways that should not change the data pattern if the null hypothesis is true. The test compares the observed statistic with values from these rearranged data sets. This works only when the null hypothesis defines a suitable group of permutations that leaves the distribution unchanged. In many interaction problems, exact permutation tests are not available, and small-sample results may be only approximate.
[2] Permutation and parametric bootstrap tests for gene ... - PMC - NIH
SEE THE IDEA
Permutation as a rearrangement
Start with elements in a fixed order or labeling.
Each element goes to exactly one target, and every target is used.
The output is a rearranged version of the same set.
Rearrangements generate comparison data when the null hypothesis allows them.
A permutation preserves the set’s elements while changing their order or assignment.
A permutation is a rearrangement of the same elements; in statistics, allowed rearrangements can be used to compare an observed result with a null distribution.
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Sources and further reading
This general lesson was generated from public source material. It explains a concept; it does not interpret a private paper or provide medical advice.