Function Repository Resource:

SignedPermutations

Source Notebook

Get all signed permutations of a list

Contributed by: Ed Pegg Jr and Jan Mangaldan

ResourceFunction["SignedPermutations"][list]

returns all the signed permutations of list.

ResourceFunction["SignedPermutations"][list,spec]

returns signed permutations of type spec.

Details

Possible settings for spec include:

All

all permutations

"Cyclic"

cyclic permutations like {{0,1,2,3},{1,2,3,0},{2,3,0,1},{3,0,1,2}}

"Even"

permutations with an even number of transpositions

"Odd"

permutations with an odd number of transpositions

"Symmetric"

same as All

Examples

Basic Examples (1)

All signed permutations of the list {1,2,3}:

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Scope (2)

All signed even permutations of the list {1,2,3}:

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All signed cyclic permutations of the list {1,2,3}:

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Applications (2)

Find the vertices of a rhombic dodecahedron with edge lengths of :

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Use ConvexHullMesh to generate the rhombic dodecahedron from its vertices:

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Properties and Relations (2)

SignedPermutations[list,All] and SignedPermutations[list,"Symmetric"] are both equivalent to SignedPermutations[list]:

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Count all signed permutations corresponding to a given list, and compare with the expressions for the explicit counts:

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$Table[2^(n - 1) n!, {n, 2, 6}]$

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$Table[2^(n - 2) n!, {n, 2, 6}]$

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$Table[2^(n - 1) n, {n, 2, 6}]$

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Possible Issues (2)

Create the 120 unit icosians that make the vertices of the 600-cell:

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$\[Phi] = RootReduce[GoldenRatio]; vecs = Join @@ (ResourceFunction["SignedPermutations"][#, "Even"] & /@ {{2, 0, 0, 0}, {1, 1, 1, 1}, {0, 1, \[Phi]^-1, \[Phi]}}/2); icosians = ResourceFunction["Quaternion"] @@ # & /@ RootReduce[vecs]; RandomSample[icosians, 8]$