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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 $k$.

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  • $\begingroup$ Is the same set of $y_i$s in a different order the same partition or a different one? $\endgroup$
    – Brendan McKay
    Jan 16, 2017 at 0:37
  • $\begingroup$ That would be the same partition. If the order is relevant, it's called a vector composition as far as I know. $\endgroup$
    – Stefan Rigger
    Jan 16, 2017 at 1:25
  • $\begingroup$ 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. $\endgroup$
    – Richard Stanley
    Jan 16, 2017 at 16:02

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Just as ordinary partitions play an important role in the theory of symmetric functions, e.g., they label basis elements, vector partitions are used in the theory of multisymmetric functions. You can look up the works on this theory by the authors I mentioned in this MO answer. Look up in particular the webpage of Emmanuel Briand.

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