Finding all partitions of a set

Starting with myList = {a, b, c, d, e, f}, here are a few solutions, in increasing order of generality:

1.

Internal`PartitionRagged[myList, #] & /@ IntegerPartitions[Length[myList]]
{{{a, b, c, d, e, f}}, {{a, b, c, d, e}, {f}}, {{a, b, c, d}, {e, f}},
 {{a, b, c, d}, {e}, {f}}, {{a, b, c}, {d, e, f}}, {{a, b, c}, {d, e}, {f}},
 {{a, b, c}, {d}, {e}, {f}}, {{a, b}, {c, d}, {e, f}},
 {{a, b}, {c, d}, {e}, {f}}, {{a, b}, {c}, {d}, {e}, {f}},
 {{a}, {b}, {c}, {d}, {e}, {f}}}

2.

Internal`PartitionRagged[myList, #] & /@ 
        Apply[Join, Permutations /@ IntegerPartitions[Length[myList]]]
{{{a, b, c, d, e, f}}, {{a, b, c, d, e}, {f}}, {{a}, {b, c, d, e, f}},
 {{a, b, c, d}, {e, f}}, {{a, b}, {c, d, e, f}}, {{a, b, c, d}, {e}, {f}},
 {{a}, {b, c, d, e}, {f}}, {{a}, {b}, {c, d, e, f}}, {{a, b, c}, {d, e, f}},
 {{a, b, c}, {d, e}, {f}}, {{a, b, c}, {d}, {e, f}}, {{a, b}, {c, d, e}, {f}},
 {{a, b}, {c}, {d, e, f}}, {{a}, {b, c, d}, {e, f}}, {{a}, {b, c}, {d, e, f}},
 {{a, b, c}, {d}, {e}, {f}}, {{a}, {b, c, d}, {e}, {f}}, {{a}, {b}, {c, d, e}, {f}},
 {{a}, {b}, {c}, {d, e, f}}, {{a, b}, {c, d}, {e, f}},
 {{a, b}, {c, d}, {e}, {f}}, {{a, b}, {c}, {d, e}, {f}},
 {{a, b}, {c}, {d}, {e, f}}, {{a}, {b, c}, {d, e}, {f}},
 {{a}, {b, c}, {d}, {e, f}}, {{a}, {b}, {c, d}, {e, f}},
 {{a, b}, {c}, {d}, {e}, {f}}, {{a}, {b, c}, {d}, {e}, {f}},
 {{a}, {b}, {c, d}, {e}, {f}}, {{a}, {b}, {c}, {d, e}, {f}},
 {{a}, {b}, {c}, {d}, {e, f}}, {{a}, {b}, {c}, {d}, {e}, {f}}}

3.

Needs["Combinatorica`"];
Short[SetPartitions[myList], 5]
{{{a, b, c, d, e, f}}, {{a}, {b, c, d, e, f}}, {{a, b}, {c, d, e, f}},
 {{a, c, d, e, f}, {b}}, {{a, b, c}, {d, e, f}}, {{a, d, e, f}, {b, c}},
 {{a, b, d, e, f}, {c}}, {{a, c}, {b, d, e, f}}, {{a, b, c, d}, {e, f}},
 {{a, e, f}, {b, c, d}}, <<184>>, {{a}, {b, d}, {c}, {e}, {f}},
 {{a}, {b, e}, {c}, {d}, {f}}, {{a}, {b, f}, {c}, {d}, {e}},
 {{a, b}, {c}, {d}, {e}, {f}}, {{a, c}, {b}, {d}, {e}, {f}},
 {{a, d}, {b}, {c}, {e}, {f}}, {{a, e}, {b}, {c}, {d}, {f}},
 {{a, f}, {b}, {c}, {d}, {e}}, {{a}, {b}, {c}, {d}, {e}, {f}}}

The output of SetPartitions[] was rather long, so I had to use Short[]. Execute SetPartitions[myList] if you want to see everything.


Based on BellList from Robert M. Dickau:

partition[{x_}] := {{{x}}}

partition[{r__, x_}] :=
  Join @@ (ReplaceList[#,
      {{b___, {S__}, a___} :> {b, {S, x}, a},
       {S__} :> {S, {x}}}
      ] & /@ partition[{r}])

This is faster than SetPartitions on shorter sets:

Needs["Combinatorica`"];
myList = {a, b, c, d, e, f};

partition[myList] ~Do~ {500} // Timing // First
SetPartitions[myList] ~Do~ {500} // Timing // First

0.405

0.843

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Combinatorics

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