Cycle Structure of the Permutation Group

Code & Zahlen

Partitions of (\mathcal S_1)

Long form Short form Young frame Pertinent cycle structure Parity Conjugacy classes to \(\mathcal A_1\) Number of elements in conjugacy class
\([1]\) \([1]\) 
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 \((\cdot)\) \(+\) (1) 1

Partitions of \(\mathcal S_2\)

Long form Short form Young frame Pertinent cycle structure Parity Conjugacy classes to \(\mathcal A_2\) Number of elements in conjugacy class
\([2, 0]\) \([2]\) 
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 \((\cdot)(\cdot)\) \(+\) (1)(2) 1
\([1, 1]\) \([1^2]\) 
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 \((\cdot\cdot)\) \(-\) (12) 1

Partitions of \(\mathcal S_3\)

Long form Short form Young frame Pertinent cycle structure Parity Conjugacy classes to \(\mathcal A_3\) Number of elements in conjugacy class
\([3, 0, 0]\) \([3]\) 
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 \((\cdot)(\cdot)(\cdot)\) \(+\) (1)(2)(3) 1
\([2, 1, 0]\) \([2, 1]\) 
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 \((\cdot\cdot)(\cdot)\) \(-\) (12)(3)(13)(2)(23)(1) 3
\([1, 1, 1]\) \([1^3]\) 
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 \((\cdot\cdot\cdot)\) \(+\) (123)(132) 2

Partitions of \(\mathcal S_4\)

Long form Short form Young frame Pertinent cycle structure Parity Conjugacy classes to \(\mathcal A_4\) Number of elements in conjugacy class
\([4, 0, 0, 0]\) \([4]\) 
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 \((\cdot)(\cdot)(\cdot)(\cdot)\) \(+\) (1)(2)(3)(4) 1
\([3, 1, 0, 0]\) \([3, 1]\) 
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 \((\cdot\cdot)(\cdot)(\cdot)\) \(-\) (12)(3)(4)(13)(2)(4)(14)(2)(3)(23)(1)(4)(24)(1)(3)(34)(1)(2) 6
\([2, 2, 0, 0]\) \([2^2]\) 
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 \((\cdot\cdot)(\cdot\cdot)\) \(+\) (12)(34)(13)(24)(14)(23) 3
\([2, 1, 1, 0]\) \([2, 1^2]\) 
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 \((\cdot\cdot\cdot)(\cdot)\) \(+\) (123)(4)(124)(3)(132)(4)(134)(2)(142)(3)(143)(2)(234)(1)(243)(1) 8
\([1, 1, 1, 1]\) \([1^4]\) 
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 \((\cdot\cdot\cdot\cdot)\) \(-\) (1234)(1243)(1324)(1342)(1423)(1432) 6

Partitions of \(\mathcal S_5\)

Long form Short form Young frame Pertinent cycle structure Parity Conjugacy classes to \(\mathcal A_5\) Number of elements in conjugacy class
\([5, 0, 0, 0, 0]\) \([5]\) 
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 \((\cdot)(\cdot)(\cdot)(\cdot)(\cdot)\) \(+\) (1)(2)(3)(4)(5) 1
\([4, 1, 0, 0, 0]\) \([4, 1]\) 
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 \((\cdot\cdot)(\cdot)(\cdot)(\cdot)\) \(-\) (12)(3)(4)(5)(13)(2)(4)(5)(14)(2)(3)(5)(15)(2)(3)(4)(23)(1)(4)(5)(24)(1)(3)(5)(25)(1)(3)(4)(34)(1)(2)(5)(35)(1)(2)(4)(45)(1)(2)(3) 10
\([3, 2, 0, 0, 0]\) \([3, 2]\) 
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 \((\cdot\cdot)(\cdot\cdot)(\cdot)\) \(+\) (12)(34)(5)(12)(35)(4)(12)(45)(3)(13)(24)(5)(13)(25)(4)(13)(45)(2)(14)(23)(5)(14)(25)(3)(14)(35)(2)(15)(23)(4)(15)(24)(3)(15)(34)(2)(23)(45)(1)(24)(35)(1)(25)(34)(1) 15
\([3, 1, 1, 0, 0]\) \([3, 1^2]\) 
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 \((\cdot\cdot\cdot)(\cdot)(\cdot)\) \(+\) (123)(4)(5)(124)(3)(5)(125)(3)(4)(132)(4)(5)(134)(2)(5)(135)(2)(4)(142)(3)(5)(143)(2)(5)(145)(2)(3)(152)(3)(4)(153)(2)(4)(154)(2)(3)(234)(1)(5)(235)(1)(4)(243)(1)(5)(245)(1)(3)(253)(1)(4)(254)(1)(3)(345)(1)(2)(354)(1)(2) 20
\([2, 2, 1, 0, 0]\) \([2^2, 1]\) 
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 \((\cdot\cdot\cdot)(\cdot\cdot)\) \(-\) (123)(45)(124)(35)(125)(34)(132)(45)(134)(25)(135)(24)(142)(35)(143)(25)(145)(23)(152)(34)(153)(24)(154)(23)(234)(15)(235)(14)(243)(15)(245)(13)(253)(14)(254)(13)(345)(12)(354)(12) 20
\([2, 1, 1, 1, 0]\) \([2, 1^3]\) 
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 \((\cdot\cdot\cdot\cdot)(\cdot)\) \(-\) (1234)(5)(1235)(4)(1243)(5)(1245)(3)(1253)(4)(1254)(3)(1324)(5)(1325)(4)(1342)(5)(1345)(2)(1352)(4)(1354)(2)(1423)(5)(1425)(3)(1432)(5)(1435)(2)(1452)(3)(1453)(2)(1523)(4)(1524)(3)(1532)(4)(1534)(2)(1542)(3)(1543)(2)(2345)(1)(2354)(1)(2435)(1)(2453)(1)(2534)(1)(2543)(1) 30
\([1, 1, 1, 1, 1]\) \([1^5]\) 
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 \((\cdot\cdot\cdot\cdot\cdot)\) \(+\) (12345)(12354)(12435)(12453)(12534)(12543)(13245)(13254)(13425)(13452)(13524)(13542)(14235)(14253)(14325)(14352)(14523)(14532)(15234)(15243)(15324)(15342)(15423)(15432) 24