# Counting the number of distinct weighted subsets

Suppose you have a finite set of items $X$. You want to assign a positive number $\mu(S)$ to each subset $S\subseteq X$ in such a way that you respect subset ordering:

$$S \subsetneq T \Rightarrow \mu(S) \lneq \mu(T).$$

In other words, you want the assignment to be monotonic.

1. There are infinitely many possible assignments of this form, but it seems that many are equivalent. Here, I consider two assignments $\mu_1$ and $\mu_2$ to be equivalent if for all subsets $S, T$, $$\mu_1(S)< \mu_1(T) \iff \mu_2(S) < \mu_2(T).$$

My first question is how many possible assignments there are, modulo this equivalence? Is there a way to parameterize or enumerate them, e.g. characterizing them by just a few of the relationships $\mu(S)\leq\mu(T)$?

2. There is a special kind of assignment which assigns a consistent weight to each element of $X$. These are the additive assignments, where $$\mu(S \sqcup T) = \mu(S) + \mu(T)$$ whenever $S$ and $T$ are disjoint subsets of $X$. My second question is how many of these there are, modulo equivalence? How can we describe them?

3. There is another wider kind of equivalence, where two assignments are equivalent if there is a permutation $\pi:X\rightarrow X$ such that $\mu_1$ on $\pi(X)$ is equivalent to $\mu_2$ on $X$ in the earlier sense. For this, it seems like there are even fewer possibilities— how can we count them?

In this second case, it seems like you need to know at least how the doubleton sets $\{x,y\}$ relate to the singleton sets, but beyond this, I'm not sure.