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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:

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

  2. 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.

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    $\begingroup$ I'm presuming you want your vectors to be nonzero. Here's the 1-dimensional case: oeis.org/A000041 $\endgroup$ Commented Mar 1, 2018 at 16:53
  • $\begingroup$ @Billy I have edited the nonzero condition part. $\endgroup$ Commented Mar 1, 2018 at 17:00
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    $\begingroup$ For some references, see arxiv.org/pdf/math/0202253.pdf and its bibliography. $\endgroup$ Commented Mar 2, 2018 at 4:10
  • $\begingroup$ @RichardStanley Dear sir, Thanks for the nice reference. $\endgroup$ Commented Mar 4, 2018 at 8:03
  • $\begingroup$ Vector partitions are also called multipartite partitions. You might search for information under that name. $\endgroup$ Commented Mar 9, 2020 at 19:46

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For rather trivial reasons, $$ 1+\sum_{(k,l)\ne (0,0)}p(k,l)x^ky^l=\prod_{(p,q)\ne(0,0)}\frac{1}{1-x^py^q} . $$ Since these numbers include, as $p(n,0)$, the one-dimensional partition numbers, you cannot really find a "closed" formula.

Similarly to how pentagonal numbers in the usual partition formulas arise in the context of Lie algebra cohomology, the denominator of this formula makes one think of the cohomology of the Lie algebra of Hamiltonian vector fields on the 2D plane, see this article and references therein.

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I think you might want to look into the Kostant partition function, and the generalization, the vector partition function. There is a lot of literature about this ,see e.g. these slides.

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