Let $k\in\mathbb{Z}_{>0}$, and $s\in\mathbb{N}$, and for $m_1,\ldots,m_k$ some nonnegative integers, consider the problem of maximizing the product $$ (1+m_1)(1+m_2)\cdots(1+m_k) $$ under the constraint $m_1+\cdots+m_k=s$.

I would like to know:

- The exact formula $M(k,s)$ for the maximal value of the product.
- A complete description of the tuples $(m_1,\ldots,m_k)$ which achieve the maximum.

Of course with the $m_i$ taking continuous real values, this is just the equality case of the arithmetic mean-geometric mean inequality and maximization calls for $m_i$'s that are "as equal to each other as possible", but forcing integer values makes this rather messy.

I can try to work my way through this, but it would feel like reinventing the wheel. A solution to 1. and 2. or pointer to the relevant literature would be appreciated.