Consider a set with elements (where all elements of are distinct). We can build ordered lists of elements by taking elements from set and using them (with no repetition) to build -tuples. With elements we can build distinct -tuples.
Let us think in terms of vectors in , where each vector represents a particular ordered list of the elements of some set with elements. If are permutations of one another, then they contain the same elements, but in a different order; consequently, for every we have
Let be an indicator function, such that if and only if are permutations of one another ( otherwise). Let be defined as
for all . We can then conclude that
The condition is necessary. Is it sufficient?! In other words, if is true, can we conclude that vectors are permutations of one another?
This is an interesting problem. Think about it!