∑Aⁿ/n! actually represents the multisets (aka bags) of elements from A. A multiset is a list, but we quotient out by permutations (hence n!) The division here

∑Aⁿ/n! actually represents the multisets (aka bags) of elements from A. A multiset is a list, but we quotient out by permutations (hence n!) The division here can be seen as an action of the permutation group (which has n! element).