Given a list of (not necessarily distinct) positive integers, $a=(a_1,...,a_n)$, can one reproduce the list (up to reversing the order, i.e. reproduce $(a_1,...,a_n)$ or $(a_n,...,a_1)$) from the underlying multiset, $\{a_1,a_2,...,a_n\}$ and the unordered pairs, $$\left\{\left\{\sum_{i=1}^k a_i,\sum_{i=k+1}^n a_i\right\}\right\}_{k\in [n]}, \quad \left\{\left\{\sum_{i=1}^k (a_i+1),\sum_{i=k+1}^n (a_i+1)\right\}\right\}_{k\in [n]}?$$
If not, how about with the information of any number of `cuts' i.e.
$$\left\{\left\{\sum_{i=1}^{k_1} a_i,\sum_{i=k_1+1}^{k_2} a_i, ..., \sum_{i=k_{l}+1}^n a_i\right\}\right\}_{k\in [n],l\in\mathbb{N}}, \quad \left\{\left\{\sum_{i=1}^{k_1} (a_i+1),\sum_{i=k_1+1}^{k_2} (a_i+1), ..., \sum_{i=k_{l}+1}^n (a_i+1)\right\}\right\}_{k\in [n],l\in\mathbb{N}}?$$
I know that if you don't also consider the collection of terms with $a_i+1$, you can't distinguish $(1,2,1,3,2)$ and $(1,3,2,1,2)$. (For more information about the cases which can't be distinguished, see https://personal.math.ubc.ca/~steph/papers/elsarticle-weightedchrom.pdf)
As an example, if the list was $(1,2,1,3,2)$ I would be given the multisets $\{1,1,2,2,3\}$, $\{\{1,8\},\{2,7\},\{3,6\},\{4,5\}\}$, and $\{\{2,12\},\{3,11\},\{5,9\},\{7,7\}\}$. (You can distinguish which set you are looking at by looking at the sum of the terms, or the minimal elements.)
EDIT : Given just a single cut, Sil found a counter example with the two lists, (1,2,1,3,1,2) and (1,2,3,1,1,2).
