Suppose I have a partition $p$ of a positive integer $n$, $p$ is defined to be a friendly partition if and only if the following hold:
- $n$ is divisible by $4$.
- The length of $p$ is $\frac{n}{2}$.
- Every part in $n$ is less than or equal to $\frac{n}{2}$.
Let's define a partition $p$ to be splittable if and only if there are two partitions $p'$ and $p''$ such that $p$ is the same partition as $p' + p''$ and the sum of $p'$ and $p''$ are the same.
Empirically, it seems to be the case that every friendly partition is balanced, although I don't have a proof.
I'm curious whether a stronger condition holds. It seems to be possible in every case to split a friendly partition $p$ into two partitions, one of which is composed solely of $\{1, 2, 3, 4, \max(p)\}$ in some combination.
I'm curious whether it is always possible to split a friendly partition such that one of its parts is composed solely of $\{1, 2, 3, 4, \max(p)\}$.
As an example, the friendly partition $1+1+1+1+1+2+3+3+3+4$ can be split where one of its parititons is $3+3+4$, which satisfies the rule above.
For a counterexample, let $n=68$ and let $p$ be the partition of $68$ given by $$ 68=(1+\cdots+1)+(5+\cdots+5)+7 $$ where $1$ has multiplicity $26$, and $5$ has multiplicity $7$.