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MathGroup Archive 1998

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Re: Permutations.

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14807] Re: [mg14771] Permutations.
  • From: BobHanlon at aol.com
  • Date: Wed, 18 Nov 1998 01:29:08 -0500
  • Sender: owner-wri-mathgroup at wolfram.com

In a message dated 11/14/98 6:19:01 AM, awhopper at hermes.net.au writes:

>For the combinations of n objects taken k at a time, (where order counts
>and there is no duplication), the function KSubsets is the one to use.
>e.g.
>      In[1]:= <<DiscreteMath`Combinatorica`
>
>      In[2]:= Table[KSubsets[{a,b,c,d}, k], {k, 4}]
>
>     Out[3]= {{{a}, {b}, {c}, {d}},
>              {{a,b}, {a,c}, {b,c}, {b,d}, {c,d}},
>              {{a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}},
>              {{a,b,c,d}}}
>
>But likewise in wanting to find all the permutation subsets (with no 
>duplication and order not counting), of a numerical or symbolic list,
>there does not seem to be a function anywhere (including the packages),
>to achieve this goal.
> 
>(By n!/(n-k)!, there will be be; 4, 12, 24, 24  permutations  taken k =
>1, 2, 3, 4  at a time, for a 4 element list).
>
>The built-in function Permutations and also LexicographicPermutations 
>(from Combinatorica) do not take a second argument, as KSubsets does,
>and so only the (n, k=n) permutations (24 in the example ) can be
>found.
>
>I would appreciate some assistance to find a way to generate all of the
>permutation subsets, in List, Table or Column form.
>

Alan,

One approach:

Needs["DiscreteMath`Combinatorica`"];

perm[theList_List, k_Integer?NonNegative] := 
	Flatten[Map[Permutations, KSubsets[theList, k]], 1] /; 
	k <= Length[theList];
perm[theList_List] := Module[{nbr = Length[theList], k},
	Table[Flatten[Map[Permutations, KSubsets[theList, k]], 1],
		{k, 0, nbr}]];

Table[perm[{a,b,c,d}, k], {k, 0, 4}]//ColumnForm

{}
{{a}, {b}, {c}, {d}}
{{a, b}, {b, a}, {a, c}, {c, a}, {a, d}, {d, a}, {b, c}, {c, b}, 
  {b, d}, {d, b}, {c, d}, {d, c}}
{{a, b, c}, {a, c, b}, {b, a, c}, {b, c, a}, {c, a, b}, {c, b, a}, 
  {a, b, d}, {a, d, b}, {b, a, d}, {b, d, a}, {d, a, b}, {d, b, a}, 
  {a, c, d}, {a, d, c}, {c, a, d}, {c, d, a}, {d, a, c}, {d, c, a}, 
  {b, c, d}, {b, d, c}, {c, b, d}, {c, d, b}, {d, b, c}, {d, c, b}} {{a,
b, c, d}, {a, b, d, c}, {a, c, b, d}, {a, c, d, b}, 
  {a, d, b, c}, {a, d, c, b}, {b, a, c, d}, {b, a, d, c}, 
  {b, c, a, d}, {b, c, d, a}, {b, d, a, c}, {b, d, c, a}, 
  {c, a, b, d}, {c, a, d, b}, {c, b, a, d}, {c, b, d, a}, 
  {c, d, a, b}, {c, d, b, a}, {d, a, b, c}, {d, a, c, b}, 
  {d, b, a, c}, {d, b, c, a}, {d, c, a, b}, {d, c, b, a}}

perm[{a,b,c,d}]//ColumnForm

{}
{{a}, {b}, {c}, {d}}
{{a, b}, {b, a}, {a, c}, {c, a}, {a, d}, {d, a}, {b, c}, {c, b}, 
  {b, d}, {d, b}, {c, d}, {d, c}}
{{a, b, c}, {a, c, b}, {b, a, c}, {b, c, a}, {c, a, b}, {c, b, a}, 
  {a, b, d}, {a, d, b}, {b, a, d}, {b, d, a}, {d, a, b}, {d, b, a}, 
  {a, c, d}, {a, d, c}, {c, a, d}, {c, d, a}, {d, a, c}, {d, c, a}, 
  {b, c, d}, {b, d, c}, {c, b, d}, {c, d, b}, {d, b, c}, {d, c, b}} {{a,
b, c, d}, {a, b, d, c}, {a, c, b, d}, {a, c, d, b}, 
  {a, d, b, c}, {a, d, c, b}, {b, a, c, d}, {b, a, d, c}, 
  {b, c, a, d}, {b, c, d, a}, {b, d, a, c}, {b, d, c, a}, 
  {c, a, b, d}, {c, a, d, b}, {c, b, a, d}, {c, b, d, a}, 
  {c, d, a, b}, {c, d, b, a}, {d, a, b, c}, {d, a, c, b}, 
  {d, b, a, c}, {d, b, c, a}, {d, c, a, b}, {d, c, b, a}}

Bob Hanlon


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