Re: S_4 elements

• To: mathgroup at smc.vnet.net
• Subject: [mg48435] Re: S_4 elements
• From: "Dr. Wolfgang Hintze" <weh at snafu.de>
• Date: Sat, 29 May 2004 03:06:58 -0400 (EDT)
• References: <c96hpf\$ii5\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Jorge,

In[29]:=
<< "DiscreteMath`Combinatorica`"

then you get all 4!=24 permutations using the function Permutation of
the package

In[30]:=
x = {1, 2, 3, 4}
y = Permutations[x]

Out[30]=
{1, 2, 3, 4}

Out[31]=
{{1, 2, 3, 4}, {1, 2, 4, 3}, {1, 3, 2, 4}, {1, 3, 4, 2},
{1, 4, 2, 3}, {1, 4, 3, 2}, {2, 1, 3, 4}, {2, 1, 4, 3},
{2, 3, 1, 4}, {2, 3, 4, 1}, {2, 4, 1, 3}, {2, 4, 3, 1},
{3, 1, 2, 4}, {3, 1, 4, 2}, {3, 2, 1, 4}, {3, 2, 4, 1},
{3, 4, 1, 2}, {3, 4, 2, 1}, {4, 1, 2, 3}, {4, 1, 3, 2},
{4, 2, 1, 3}, {4, 2, 3, 1}, {4, 3, 1, 2}, {4, 3, 2, 1}}

The cycle structure of the permutations can be obtained using the
function ToCycles contained in the package, ie.

In[45]:=
ToCycles /@ y

Out[45]=
{{{1}, {2}, {3}, {4}}, {{1}, {2}, {4, 3}},
{{1}, {3, 2}, {4}}, {{1}, {3, 4, 2}}, {{1}, {4, 3, 2}},
{{1}, {4, 2}, {3}}, {{2, 1}, {3}, {4}},
{{2, 1}, {4, 3}}, {{2, 3, 1}, {4}}, {{2, 3, 4, 1}},
{{2, 4, 3, 1}}, {{2, 4, 1}, {3}}, {{3, 2, 1}, {4}},
{{3, 4, 2, 1}}, {{3, 1}, {2}, {4}}, {{3, 4, 1}, {2}},
{{3, 1}, {4, 2}}, {{3, 2, 4, 1}}, {{4, 3, 2, 1}},
{{4, 2, 1}, {3}}, {{4, 3, 1}, {2}}, {{4, 1}, {2}, {3}},
{{4, 2, 3, 1}}, {{4, 1}, {3, 2}}}

Or, if you like to drop trivial cycles of length 1,

define

In[38]:=
toEssentialCycles[p_] := Select[ToCycles[p], Length[#1] > 1 & ]

and get

In[43]:=
toEssentialCycles /@ y

Out[43]=
{{}, {{4, 3}}, {{3, 2}}, {{3, 4, 2}}, {{4, 3, 2}},
{{4, 2}}, {{2, 1}}, {{2, 1}, {4, 3}}, {{2, 3, 1}},
{{2, 3, 4, 1}}, {{2, 4, 3, 1}}, {{2, 4, 1}},
{{3, 2, 1}}, {{3, 4, 2, 1}}, {{3, 1}}, {{3, 4, 1}},
{{3, 1}, {4, 2}}, {{3, 2, 4, 1}}, {{4, 3, 2, 1}},
{{4, 2, 1}}, {{4, 3, 1}}, {{4, 1}}, {{4, 2, 3, 1}},
{{4, 1}, {3, 2}}}

Regards,
Wolfgang

Jorge Luis Llanio wrote:

> Hi everybody in the list!
>
> please, I need the listing of the symmetric S_4 group elements, ex.:
>
> (1)(2)(3)(4);  (1234); (12)(34), etc      a total of 4! = 24 elements
>
>
> Thank you very much in advance,   Jorge
>
>

```

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