I am currently reading a paper (Iterated Binomial Coefficients by S.W. Golomb, The American Mathematical Monthly, 1980, 719-727) that makes use of the "optimality principle" in a couple of proofs. One theorem in which they rely on it is as follows:
Let $n$ be a positive integer and define $g(n) = \max a_{1} \cdot a_{2} \cdots a_{k}$, where the product is maximized over all partitions of $n$ into positive integers, $n = a_{1} + a_{2} + \cdots + a_{k}$. Then for $n > 1$, $$g(n)= \begin{cases} 3 \cdot 3 \cdots 3 \cdot 3 = 3^{n/3} & \text{if } n \equiv 0 \mod 3 \\ 3 \cdot 3 \cdots 3 \cdot 4 = 4 \cdot 3^{(n-4)/3}& \text{if } n \equiv 1 \mod 3 \\ 3 \cdot 3 \cdots 3 \cdot 2 = 2 \cdot 3^{(n-2)/3}& \text{if } n \equiv 2 \mod 3 \end{cases} $$
In the proof, they handle the case for $n \leq 4$ by exhaustive search, but for $n > 5$ they assert that "it is clear that the "optimality principle" applies, i.e., $g(n) = \max_{1 < a < n} a \cdot g(n-a)$."
My question is, what exactly is this "optimality principle"? How is it so clear that the optimality principle applies for the case when $n > 5$ Is there a more formal definition? If so, is there a text that I could reference to help my understanding.