Suppose we have $n$ variables $a_1 \leq a_2 \leq \dots \leq a_n$. If the values of the variables are not known, in how many ways could $\{a_i + a_j\}_{\{i,j\} \in {[n] \choose 2}}$ be ordered using $<$ and $=$?
For example, if $n=4$, we would have $a_1, a_2, a_3, a_4$. If we assign $a_1 = 0, a_2 = 1, a_3 = 5, a_4 = 12$, then $a_1+a_2 < a_1 + a_3 < a_2 + a_3 < a_1 + a_4 < a_2 + a_4 < a_3 + a_4$.
However, if we assign $a_1 = 1, a_2 = 4, a_3 = 5, a_4 = 6$, then $a_1 + a_2 < a_1+a_3 < a_1 + a_4 < a_2 + a_3 < a_2 + a_4 < a_3 + a_4$, which is a different ordering of $\{a_i + a_j\}$. This means that for $n=4$, we have at least two orderings of $\{a_i + a_j\}$. Of course there are many more.
For $n$ variables, how many of these orderings of $\{a_i + a_j\}$ are possible? I'd like $a_{i_1} + a_{j_1} < a_{i_2} + a_{j_2}$ to be distinct from $a_{i_1} + a_{j_1} = a_{i_2} + a_{j_2}$. Would anything preserving the natural partial ordering work?
I'd appreciate any help or suggestions. Thank you.