We say that $A=\sum_{i=1}^a a_i$ and $B=\sum_{j=1}^b b_j$ is a *unique partition* of $A$ and $B$ if there is no other way to partition the $a+b$ numbers into two parts that sum to $A$ and $B$. This is meant to also imply that $a_i\ne b_j$, but we allow $a_i=a_{i'}$. For example, $6=3+3$ and $53=13\times 4+1$ is a unique partition of $6$ and $53$, and so is $6=3+3$ and $53=4\times 13+1$. Let

$mup(A,B)=\max \{a+b\mid$ there is a unique partition $A=\sum_{i=1}^a a_i$ and $B=\sum_{j=1}^b b_j\}$

so the (bigger) above partition shows that $mup(6,53)\ge 16$.
Has this been studied? Is there a simple formula for it?
I know some bounds for special cases, but no general formula.
Let me also mention the following "law:" $mup(n+\nu(c),c)=mup(n,c)+1$ for large enough $n$, where $\nu(c)$ is the smallest natural that does *not* divide $c$. The bounds

$\frac n{\nu(c)}-O(1)\le mup(n,c)\le \frac n{\nu(c)}+O(c)$

are easy to see to hold.