Cycle Structure of the Permutation Group
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 |
---|---|---|---|---|---|---|
\([1]\) | \([1]\) | # |
\((\cdot)\) | \(+\) | (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 |
---|---|---|---|---|---|---|
\([2, 0]\) | \([2]\) | ## |
\((\cdot)(\cdot)\) | \(+\) | (1)(2) | 1 |
\([1, 1]\) | \([1^2]\) | # # |
\((\cdot\cdot)\) | \(-\) | (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 |
---|---|---|---|---|---|---|
\([3, 0, 0]\) | \([3]\) | ### |
\((\cdot)(\cdot)(\cdot)\) | \(+\) | (1)(2)(3) | 1 |
\([2, 1, 0]\) | \([2, 1]\) | ## # |
\((\cdot\cdot)(\cdot)\) | \(-\) | (12)(3)(13)(2)(23)(1) | 3 |
\([1, 1, 1]\) | \([1^3]\) | # # # |
\((\cdot\cdot\cdot)\) | \(+\) | (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 |
---|---|---|---|---|---|---|
\([4, 0, 0, 0]\) | \([4]\) | #### |
\((\cdot)(\cdot)(\cdot)(\cdot)\) | \(+\) | (1)(2)(3)(4) | 1 |
\([3, 1, 0, 0]\) | \([3, 1]\) | ### # |
\((\cdot\cdot)(\cdot)(\cdot)\) | \(-\) | (12)(3)(4)(13)(2)(4)(14)(2)(3)(23)(1)(4)(24)(1)(3)(34)(1)(2) | 6 |
\([2, 2, 0, 0]\) | \([2^2]\) | ## ## |
\((\cdot\cdot)(\cdot\cdot)\) | \(+\) | (12)(34)(13)(24)(14)(23) | 3 |
\([2, 1, 1, 0]\) | \([2, 1^2]\) | ## # # |
\((\cdot\cdot\cdot)(\cdot)\) | \(+\) | (123)(4)(124)(3)(132)(4)(134)(2)(142)(3)(143)(2)(234)(1)(243)(1) | 8 |
\([1, 1, 1, 1]\) | \([1^4]\) | # # # # |
\((\cdot\cdot\cdot\cdot)\) | \(-\) | (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 |
---|---|---|---|---|---|---|
\([5, 0, 0, 0, 0]\) | \([5]\) | ##### |
\((\cdot)(\cdot)(\cdot)(\cdot)(\cdot)\) | \(+\) | (1)(2)(3)(4)(5) | 1 |
\([4, 1, 0, 0, 0]\) | \([4, 1]\) | #### # |
\((\cdot\cdot)(\cdot)(\cdot)(\cdot)\) | \(-\) | (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 |
\([3, 2, 0, 0, 0]\) | \([3, 2]\) | ### ## |
\((\cdot\cdot)(\cdot\cdot)(\cdot)\) | \(+\) | (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 |
\([3, 1, 1, 0, 0]\) | \([3, 1^2]\) | ### # # |
\((\cdot\cdot\cdot)(\cdot)(\cdot)\) | \(+\) | (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 |
\([2, 2, 1, 0, 0]\) | \([2^2, 1]\) | ## ## # |
\((\cdot\cdot\cdot)(\cdot\cdot)\) | \(-\) | (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 |
\([2, 1, 1, 1, 0]\) | \([2, 1^3]\) | ## # # # |
\((\cdot\cdot\cdot\cdot)(\cdot)\) | \(-\) | (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 |
\([1, 1, 1, 1, 1]\) | \([1^5]\) | # # # # # |
\((\cdot\cdot\cdot\cdot\cdot)\) | \(+\) | (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 |