# How can I more quickly get the interchangeable variables in a function?

Let’s say I have an expression like `a b + c d e`. This is equal to: `a b + c d e`, `a b + c e d`, `a b + d c e`, `a b + d e c`, `a b + e c d`, `a b + e d c`, `b a + c d e`, `b a + c e d`, `b a + d c e`, `b a + d e c`, `b a + e c d`, and `b a + e d c`

As you can see, swapping `a` and `b` will always result in an expression equal to the original expression and swapping `c`, `d`, and `e` will always result in an expression equal to the original expression.

The following code is too slow for more complicated expressions. Is there a speedier way to get Mathematica to give me all equivalence classes of swappable variables such that `f[a b + c d e] == {{a, b}, {c, d, e}}` for some `f`?

``SwapVariables[expr_, variable1_, variable2_] := expr /. variable1 -> replacedInSwapVariablesFunction /. variable2 -> variable1 /. replacedInSwapVariablesFunction -> variable2;  VariablesIn[expr_] := Integrate`getAllVariables[{expr}, {}];  SwappableVariablesIn[expr_] := (   vars = VariablesIn[expr];   originalVars = vars;   results = {};   While[Length[vars] > 0, (     var = First[vars];     vars = Rest[vars];     swappable = Map[TrueQ[ForAll[originalVars, SwapVariables[expr, var, #] == expr]] &, vars];     results = Append[results, Prepend[Pick[vars, swappable], var]];     vars = Pick[vars, swappable, False];   )];   results );  SwappableVariablesIn[a b + c d e]  (* {{a,b},{c,d,e}} *) ``

A function which can be used for a complicated expression to test timing:

``DetNByN[n_] := Det[Table[Table[Indexed[x, {i, j}], {j, 1, n}], {i, 1, n}]];  First[Timing[SwappableVariablesIn[DetNByN[6]]]]  (* 10.7118 *) ``