Permutations
Permutations[list]
generates a list of all possible permutations of the elements in list.
Permutations[list,n]
gives all permutations containing at most n elements.
Permutations[list,{n}]
gives all permutations containing exactly n elements.
Details
- There are n! permutations of a list of n distinct elements.
- Repeated elements are treated as identical. »
- The object list need not have head List.
- Permutations[list] is effectively equivalent to Permutations[list,{Length[list]}].
- Permutations[list,{nmin,nmax}] gives permutations of list between nmin and nmax elements. Permutations[list,{nmin,nmax,dn}] uses step dn.
- Permutations[list,All] is equivalent to Permutations[list,{0,Length[list]}].
Examples
open all close allBasic Examples (2)
Scope (4)
Repeated elements are treated as identical:
Permutations[{a, a, b}]Use any expressions as elements:
Permutations[{x, x ^ 2, x + 1}]Get permutations of all lengths, shortest ones first:
Permutations[Range[3], All]Get even-length permutations, longest ones first:
Permutations[Range[4], {4, 0, -2}]Generalizations & Extensions (1)
Properties & Relations (4)
The number of length-n permutations of a length-n list of distinct elements is n!:
Length[Permutations[Range[5]]]5!The number of length-r permutations of a length-n list of distinct elements is FactorialPower[n,r]:
Length[Permutations[Range[5], {3}]]FactorialPower[5, 3]A permutation that leaves no element invariant is called a derangement:
derangementQ[perm_] := Inner[UnsameQ, perm, Sort[perm], And]The number of derangements of n distinct elements is Subfactorial[n]:
Count[Permutations[Range[5]], _ ? derangementQ]Subfactorial[5]If the input list is in the order given by Sort, so are its length-r permutations:
Sort[Permutations[{a, b, c, d}, {3}]] === Permutations[{a, b, c, d}, {3}]
An Elementary Introduction to the Wolfram LanguageText
Wolfram Research (1988), Permutations, Wolfram Language function, https://reference.wolfram.com/language/ref/Permutations.html (updated 2007).
CMS
Wolfram Language. 1988. "Permutations." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/Permutations.html.
APA
Wolfram Language. (1988). Permutations. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Permutations.html
BibTeX
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