It could be ambiguous question.
Is there a notion for a partition function including "multiplicities"?
For example, the above "multiplicities" means that
$$3=1+1+1={\color{red}{1+2}}={\color{red}{2+1}}$$
counting $1+2$ and $2+1$ are distinguished.
If anyone knows the relevant concepts, can you give me a reference or a textbook?
Thank you!
When order matters, the term used is Compositions rather than Partitions. As a first step you could check out the Wikipedia entry for these.