Combinations in combinatorics have a special meaning. Accordingly, you can calculate the combinations for more dimensions. Note that these always contain the indices of diagonal elements. See threads on StackOverflow here (combinations)here (combinations) and here (combinations with repetition)here (combinations with repetition).
This solution is the function version of the answer provided by Yaroslav Bulatov (link above):
Combinations[elem_List] := Combinations[elem, All]; Combinations[elem_List, All | Full] := Combinations[elem, Length@Union@elem]; Combinations[elem_List, n_Integer] := Module[{coef2vars, coefs, set}, set = Union@elem; coef2vars[list_] := Join @@ (MapIndexed[Table[set[[First@#2]], {#1}] &, list]); coefs = Flatten[Permutations /@ IntegerPartitions[n, {Length@set}, Range[0, n]], 1]; Sort@(coef2vars /@ coefs) ]; Combinations[{1, 2, 3, 4}, 3] {{1, 1, 1}, {1, 1, 2}, {1, 1, 3}, {1, 1, 4}, {1, 2, 2}, {1, 2, 3}, {1, 2, 4}, {1, 3, 3}, {1, 3, 4}, {1, 4, 4}, {2, 2, 2}, {2, 2, 3}, {2, 2, 4}, {2, 3, 3}, {2, 3, 4}, {2, 4, 4}, {3, 3, 3}, {3, 3, 4}, {3, 4, 4}, {4, 4, 4}}
