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sorting - How to get all possible orderings (equal ranks allowed) for given set of N objects?


Hope there is a solution besides the tedious generation of nested loops. (Trying to avoid reinventing the wheel.)


Here is an example with N=3 objects. There are 13 needed orderings (first {} means first place, second {} means second place, ...):


1. {a}, {b}, {c}
2. {a}, {c}, {b}
3. {a}, {b, c}
4. {b}, {a}, {c}
5. {b}, {c}, {a}
6. {b}, {a, c}
7. {c}, {a}, {b}

8. {c}, {b}, {a}
9. {c}, {a, b}
10. {a, b}, {c}
11. {a, c}, {b}
12. {b, c}, {a}
13. {a, b, c}

How can I get all such orderings for a given N?


UPD: I also wonder how to get same orderings in binary relations notation, i.e., considering orderings as sets of ordered pairs (also neglecting here such pairs as (a,a), (b,b), ... since they don't make further difference). I found out that this notation is much easier way to further operating with rankings in Mathematica. Here are the above 13 orderings in new notation:


1. {(a,b), (a,c), (b,c)}

2. {(a,c), (a,b), (c,b)}
3. {(a,b), (a,c), (b,c) (c,b)}
4. {(b,a), (b,c), (a,c)}
5. {(b,c), (b,a), (c,a)}
6. {(b,a), (b,c), (a,c), (c,a)}
7. {(c,a), (c,b), (a,b)}
8. {(c,b), (c,a), (b,a)}
9. {(c,a), (c,b), (a,b), (b,a)}
10. {(a,b), (b,a), (a,c), (b,c)}
11. {(a,c), (c,a), (a,b), (c,b)}

12. {(b,c), (c,b), (b,a), (c,a)}
13. {(a,b), (b,a), (a,c), (c,a), (b,c), (c,b)}

Answer



You can use ReplaceList with a helper function which has the Orderless attribute:


ClearAll[f]; SetAttributes[f, Orderless];

ReplaceList[f[a, b, c], f[a___, b___, c___] :> {{a}, {b}, {c}}] //
   DeleteCases[#, {}, -1] & // Union // Column

enter image description here



The DeleteCases and Union are required because the output from ReplaceList includes the empty list {} as a distinct entity.


For an arbitrary input list the pattern has to be constructed with the appropriate number of arguments:


orderings[x_] := Module[{f},
  SetAttributes[f, Orderless];
  ReplaceList[f @@ x, With[{s = Table[Unique[], {Length@x}]},
      Pattern[#, ___] & /@ f @@ s :> Evaluate[Thread[{s}]]]] //
    DeleteCases[#, {}, -1] & // Union]

Style[orderings[{1, 2, 3, 4}], Small]


enter image description here


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