What is this function

Question

Let $\left(\delta_{i,j}\right)_{5\times5}$ be a matrix with $\delta_{i,i}=1$ and $\delta_{i,j}=\delta_{j,i}=1$ or $0$ when $i\ne j$ and $\delta_{i}=\sum_{j=1}^{5}\delta_{i,j}$. Let $$ f(k):=\max_{\delta_{1}+\delta_{2}+\delta_{3}+\delta_{4}+\delta_{5}=k}\delta_{1}\delta_{2}\delta_{3}\delta_{4}\delta_{5} $$ for $5\le k\le25$, then what is the combinatorial expression of $f(k)$ as a function of $k$?

Answer 1

Your $\delta_i$ is just how many of $1..5$ are equal to $i$, i.e., 1 if $1 \le i \le 5$. The maximum of the products of $\delta_i$ is thus one (if all 5 give one), and that can happen only if their sum is 5 (five of those can't ever sum more than 5). Thus $f(n) = \delta_{n, 5}$.