Let $S=\{a,b,c,d,...\}$. Let $P_n=(abc+abd+acd+...+ab+ac+ad+...a+b+c+d...)^n$. In addition, there's the condition that for all variables, $x^n=x$ (maybe it'll be easier without this?). Is there something like the multinomial theorem to expand out $P_n$?
Edit: Here's what I have so far: For any $n$, the unique terms in the expanded product is the set of terms $a^{k_1}b^{k_2}c^{k_3}...$, such that $k_1,k_2,k_3,... \le n$ (you can have at most $n$ multiplicands in the product), and that $n \le k_1+k_2+k_3+... \le 3n$ (at minimum, there are $n$ length-1 multiplicands in the product, and at most, there are $n$ length-3 multiplicands in the product). For any such term $a^{k_1}b^{k_2}c^{k_3}...$, to find the number of terms that make it up, there must be $k_1$ terms that have $a$, $k_2$ terms that includes $b$, $k_3$ that includes c, and so on.
Edit 2: I realize that this can be done recursively --- for a given term, remove one of the original terms ($abc, abd, ...$), and find the composition of that smaller term,