This is a follow up to a question about determining the set of values from a set of sums.
We consider the following setup:
Consider a vector $A$ (which you will not see) of $n$ positive integers. You are given the set of sums of the (contiguously indexed) subvectors of $A$. For example, say
$$A = (3,2,1,2)$$
The subvectors are $(3),(2),(1),(2), (3,2), (2,1), (1,2),(3,2,1), (2,1,2),(3,2,1,2)$. We would be given the sums $\{1, 2, 3, 5, 6, 8\}$. Let us call this set of sums $f(A)$.
It turns out it is not always possible to uniquely determine the set of integers in $A$ from the pair $(f(A), n)$? For example:
$$A = (1,1,3)$$ and $$B = (1,2,2)$$
both give the same set of sums $f(A) = f(B) = \{1,2,3,4,5\}$.
However in some cases it is possible. For example, if $n=3$ and $f(C) = \{1,2,3,5,6\}$ then we know that $\operatorname{set}(C) = \{1, 2, 3\}$.
This raises the following question:
Is it possible to characterise which pairs $(f(X), n)$ uniquely determine the set of values in the array $X$?