Partition of $2r$ in $k$ summands of given product

Question

Given the integers $r$, $k$ and $P$, is there a unique partition of $2r$ in $k$ summands the product of which is the closest to $P$? If yes, how can we find it?

Answer 1

Uniqueness fails for two versions of the question.

For partitions with product equal to $P$, the smallest counterexample is $r = 7$, $k = 3$, and $P = 72$ as $(6,6,2)$ and $(8,3,3)$ are 3-part partitions of 14 with the same product, 72.

For partitions with product closest to but not equal to $P$, there is already a problem at $r = 3$ and $k = 3$: the 3-part partitions of 6, namely $(2,2,2),(3,2,1), (4,1,1)$, have products $4, 6, 8$ so that there is not a unique partition whose product is closest to $P = 5$ or $P = 7$.

By the way, the product of the parts of a partition is sometimes called the norm and was the topic of a survey by Schneider and Sills published last year in the journal Integers.