Re: Listing the partitions of a set

• To: mathgroup at smc.vnet.net
• Subject: [mg65311] Re: Listing the partitions of a set
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Sat, 25 Mar 2006 05:17:48 -0500 (EST)
• Organization: The Open University, Milton Keynes, UK
• References: <e0022d\$plp\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Richard Palmer wrote:
> Is there a simple way to list the partitions of a set?  For example, there
> are 15 partitions on a set of 4 elements.  {{{1, 2, 3, 4}}, {1, {2, 3, 4}},
> {{1, 3, 4}, 2}, {{1, 2, 4}, 3}, {{1, 2, 3},    4}, {{1, 2}, {3, 4}}, {{1,
> 3}, {2, 4}}, {{1, 4}, {2, 3}}, {1, 2, {3, 4}}, {1, 3, {2, 4}}, {1, 4, {2,
> 3}}, {2, 3, {1, 4}}, {2,    4, {1, 3}}, {3, 4, {1, 2}}, {1, 2, 3, 4}} .
>
>
Hi Richard,

You can use the *SetPartitions* function from the *Combinatorica* package:

In[1]:=
Needs["DiscreteMath`Combinatorica`"]

In[2]:=
SetPartitions[4]

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

In[3]:=
Length[%]

Out[3]=
15

Best regards,
/J.M.

```

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