For a finite set $S$ of $n$ elements, say a weight function is a function $f \colon S \to \mathbb{R}$. For any subset $T \subseteq S$, define $f(T) = \sum _{x \in T} f(x)$. Define two weight functions $f_1, f_2$ on the same set $S$ to be *equivalent* if $f_1(T_1) \leq f_1(T_2) \Leftrightarrow f_2(T_1) \leq f_2(T_2)$ for all sets $T_1, T_2 \subseteq S$.

How many non-equivalent weight functions are there on a set of $n$ elements, asymptotically as a function of $n$? It follows from a result of Frank and Tardos (Thm 3.3) that there are at most $2^{O(n^4)}$ distinct weight functions, but I am interested to know what the optimal degree is of the polynomial in the exponent.