# References on Vector Partitions

I'm looking for references on vector partitions, which generalize integer partitions. A vector partition of a nonzero vector $x \in \mathbb{N}^{n}\setminus\{\bf0\}$ is a sequence of vectors $y_1,\dots,y_k \in \mathbb{N}^{n}\setminus\{\bf 0\}$ such that $$y_1 + \dots + y_k = x.$$ Vector partitions are closely related to partitions of multisets. I'm looking for references on properties / combinatorial identities of vector partitions, unfortunately the results of my own efforts searching on the internet have so far been rather sparse.

The question I'm interested in is how the number of vector partitions of $x$ into exactly $k$ parts is related to the number of vector partitions of $x$ into parts where the largest part has norm $k$.

• Is the same set of $y_i$s in a different order the same partition or a different one? – Brendan McKay Jan 16 '17 at 0:37
• That would be the same partition. If the order is relevant, it's called a vector composition as far as I know. – Stefan Rigger Jan 16 '17 at 1:25
• Kostant's partition function is a special case of vector partitions. See arxiv.org/pdf/1101.0388v1.pdf and the reference [BV] of this paper. – Richard Stanley Jan 16 '17 at 16:02