I am interested in partitioning a vector with nonnegative integer entries into a sum of vectors with nonnegative integral entries. For example the partitions like (2,2) = (1,1)+(1,1) = (2,0)+(0,2) = (1,0)+(0,1)+(1,1) = ... .

I have the following questions:

Given a vector $\textbf{b} \ne \textbf{0}$ whether the number of such partitions is known in the literature?

What is the combinatorial significance of this number?

Kindly share your views on these questions and thanks for your valuable time.

Have a good day.