Has anyone heard of a function $f$: $\mathbb{Z}^{n^+} \to \mathbb{Z}^{n^+}$, such that if $S= \{{a_1,a_2,...,a_n}\},f(S):=\sum\limits_{s_k \subset S, |s_k|=n-1,s_k \not= s_j} f(s_k)$ and $f(n)=n$? For example, $f(1,2,3)=f(1,2)+f(2,3)+f(3,1)=f(1)+f(2)+f(2)+f(3)+f(3)+f(1)=12.$ It is obvious that with k distinct elements we have $f(a_1,a_2,...,a_k)=(k-1)!\sum\limits_{n=1}^k a_n$ but it gets trickier with more variance. $f(1,1,2,3)=f(1,2,3)+f(1,1,2)+f(1,1,3)=12+1+3+1+4=21$.
I'm just having trouble pinning this function down. I hope you can forgive my slightly abusive notation.