# What is the correct technical term for this generalization of an integer partition?

Given a vector $v$ with non-negative integer coordinates, is there a technical term for an unordered tuple of vectors $(v_1,\dots, v_k)$ with non-negative integer coordinates such that

$v_1+\dots+v_k = v$?

I would have liked to have called it a Vector Partition of $v$, but that term seems to be used for a decomposition a vector as a linear combination with non-negative integral coefficients of a fixed set of vectors.

An authoritative reference for the usage would also be welcome.

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What do you mean by "unordered tuple"? – Jesko Hüttenhain Jan 29 '13 at 7:02
I mean that the order in which the summands $v_1,\dots,v_k$ are taken does not matter, but the number of times each $v_i$ occurs does matter (this is akin to integer partitions). – Amritanshu Prasad Jan 29 '13 at 8:05
Actually, I believe that an integer partition is exactly what you want. If the order does not matter, then you can as well put them in a descending order. If the order would matter, I would consider multiindices. – Jesko Hüttenhain Jan 29 '13 at 9:01
How is your definition different from "a decomposition a vector as a linear combination with non-negative integral coefficients of any nonzero vectors with non-negative integral entries"? (I take it you forbid a zero vector among the $v_i$, since that would cause infitely many possibilities. Or do you fix the values of $k$ beforehand?) In other words, isn't this a special case of the Vector Partition problem with a specific infinite fixed set of vectors? – Marc van Leeuwen Jan 29 '13 at 12:39
@Marc Yes, indeed, it is a special case. I want to know if this special case (where all non-zero integer vectors are allowed) has a special name. Do I gather correctly from your comment that one cannot do better than "Vector Partition"? – Amritanshu Prasad Jan 29 '13 at 16:49