  Let $A\subset \mathbb{N}^{n}$ be an $r$-wise $s$-union family, that is, a
family of sequences with $n$ components of non-negative integers such that for
any $r$ sequences in $A$ the total sum of the maximum of each component in
those sequences is at most $s$. We determine the maximum size of $A$ and its
unique extremal configuration provided (i) $n$ is sufficiently large for fixed
$r$ and $s$, or (ii) $n=r+1$.
