How can I obtain all partitions of a list in Mathematica?

By partition of a list, I mean a set of subsets of the list elements such that the intersection of any distinct pair of subsets is empty, and the union of all subsets is equal to the original list.

For example, if my input list is {1,π,x} then I'd like a function that returns

{ {{1},{π},{x}}, {{1,π},{x}}, {{1,x},{π}}, {{1},{x,π}}, {{1,π,x}} }

Solution

Using adapted code from Link

BellList[1] = {{{1}}};
BellList[n_Integer?Positive] := Join @@
  (ReplaceList[#,
    {{b___, {S__}, a___} :> {b, {S, n}, a},
     {S__} :> {S, {n}}}
   ] & /@ BellList[n - 1])

s = {a, b, c, d, e};

bell = BellList@Length@s /. n_Integer :> s[[n]]

Or, unsurprisingly, the Combinatorica package has this function (SetPartitions) already!

Needs["Combinatorica`"]

comb = SetPartitions[{a, b, c, d, e}]

check that they both return the same result (but in different orders)

Complement[bell, comb] == {}
Sort@bell == Sort@comb
(* Both of the above return True *)