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,
first load the appropriate package
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
>
>