# 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
$[1]$ $[1]$
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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