PermutationProduct

The product of permutations, applying the left factor first. $(p\,q)(i) = q(p(i))$.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguagePermutationProduct✓↗

PermutationProduct(p, q, ...)the permutation that applies p, then q, and so on

SageMathenumeratio_permutation_product(p, q, ...)

Typed signatures

Type	Package	Applies
((expression<Cycles> | list<integer> | permutation)*) -> expression<Cycles> | list<integer> | permutation	combinatoricsreplaces groupalgebra	always
((expression<Cycles> | list<integer>)*) -> expression<Cycles> | list<integer>	groupalgebra	always

Domain: Permutations

Details

• Either notation, as Permute and InversePermutation read them: a one-line word {v1, ..., vn} or Cycles, or a Permutation
• The result is Cycles when any argument is, else a one-line word (a Permutation when given one); shorter words are padded with fixed points
• Wolfram multiplies left to right, so PermutationProduct(p, q) is not $p \circ q$; it is $q \circ p$
• With no arguments it is the identity, Cycles({})

Examples

Basic #

$1 \to 2 \to 1$, $2 \to 3 \to 3$, $3 \to 1 \to 2$

live ↗

In$\mathrm{PermutationProduct}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack))$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack2, 3\bigr\rbrack\bigr\rbrack)$

Known value · wolframscript 15.0

Scope #

swapping the factors gives a different permutation

live ↗

In$\mathrm{PermutationProduct}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack\bigr\rbrack))$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3\bigr\rbrack\bigr\rbrack)$

Known value · wolframscript 15.0

live ↗

In$\mathrm{PermutationProduct}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack\bigr\rbrack))$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack)$

Known value · wolframscript 15.0

two one-line words give a one-line word

live ↗

In$\mathrm{PermutationProduct}(\bigl\lbrack2, 3, 1\bigr\rbrack, \bigl\lbrack2, 1, 3\bigr\rbrack)$Out$\bigl\lbrack1, 3, 2\bigr\rbrack$

Known value · wolframscript 15.0

the shorter word is padded with fixed points

live ↗

In$\mathrm{PermutationProduct}(\bigl\lbrack2, 1\bigr\rbrack, \bigl\lbrack1, 3, 2\bigr\rbrack)$Out$\bigl\lbrack3, 1, 2\bigr\rbrack$

Known value · wolframscript 15.0

any Cycles argument makes the result Cycles

live ↗

In$\mathrm{PermutationProduct}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack), \bigl\lbrack2, 3, 1\bigr\rbrack)$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3\bigr\rbrack\bigr\rbrack)$

Known value · wolframscript 15.0

Permutation values give a Permutation

live ↗

In$\mathrm{PermutationProduct}({\operatorname{Permutation}}([2, 3, 1]), {\operatorname{Permutation}}([2, 1, 3]))$Out${\operatorname{Permutation}}([1, 3, 2])$

Known value · wolframscript 15.0, on the one-line word

the smaller degree is padded with fixed points

live ↗

In$\mathrm{PermutationProduct}({\operatorname{Permutation}}([2, 1]), {\operatorname{Permutation}}([1, 3, 2]))$Out${\operatorname{Permutation}}([3, 1, 2])$

Known value · wolframscript 15.0, on the one-line word

any Cycles argument makes the result Cycles

live ↗

In$\mathrm{PermutationProduct}({\operatorname{Permutation}}([2, 3, 1]), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack))$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack2, 3\bigr\rbrack\bigr\rbrack)$

Known value · wolframscript 15.0, on the one-line word

Properties #

the empty product is the identity

live ↗

In$\mathrm{PermutationProduct}()$Out$\mathrm{Cycles}(\bigl\lbrack\bigr\rbrack)$

Known value · wolframscript 15.0

About

Examples	10
Checked against	Wolfram Language 10 agreeSageMath 9 agree
Crosswalk	Wolfram LanguageSageMath ×2

Implementation

mappedsagepackages/oracle/src/mappings.ts

Sage multiplies left to right by default, applying the left factor first, as Wolfram and ours do.

mappedsagepackages/oracle/src/mappings.ts

See also: PermutationPower, InversePermutation, Permute