  In this paper, we study the "sum composition problem" between two lists $A$
and $B$ of positive integers. We start by saying that $B$ is "sum composition"
of $A$ when there exists an ordered $m$-partition $[A_1,\ldots,A_m]$ of $A$
where $m$ is the length of $B$ and the sum of each part $A_k$ is equal to the
corresponding part of $B$. Then, we consider the following two problems: $i)$
the "exhaustive problem", consisting in the generation of all partitions of $A$
for which $B$ is sum composition of $A$, and $ii)$ the "existential problem",
consisting in the verification of the existence of a partition of $A$ for which
$B$ is sum composition of $A$. Starting from some general properties of the sum
compositions, we present a first algorithm solving the exhaustive problem and
then a second algorithm solving the existential problem. We also provide proofs
of correctness and experimental analysis for assessing the quality of the
proposed solutions along with a comparison with related works.
