The number 3 can be summed by $1 + 1 + 1$ or $1 + 2$.
The set of all multisets of summands would be $\left\{\left\{1,1,1\right\},\left\{1,2\right\}\right\}$
Is there a general term for this kind of comprehensive set of possible decompositions?
Maybe no specific word for summands exists, but there might be a word for this concept for factors? E.g. 12 is 2×6, 2×3×2, 3×4, etc.
The partition function $p(n)$ counts the number of partitions $\lambda$ of $n$ (denoted by $\lambda\vdash n$), i.e. sequences $\lambda$ such that $$ 1\leq\lambda_1\leq\lambda_2\leq\ldots, \quad \sum_i\lambda_i=n. $$ The ordinary generating function for $p(n)$ is $\prod_{k=1}^{\infty}\frac{1}{1-x^k}$, which is closely related to the modular discriminant or the Dedekind eta function. Euler noted a nice recurrence (Euler's pentagonal number theorem), Hardy and Ramanujan applied the circle method to derive asymptotics, Ramanujan noted congruences, etc., etc. This function has a long and storied history and will continue to intrigue mathematicians for years to come. https://en.wikipedia.org/wiki/Partition_(number_theory)