I would like to find a simplification of the expression \begin{equation} \log{\frac{(x+y+z)!}{x! y! z!}} \end{equation} that is linear with respect to $x, y$, and $z$. Does such an expression exist?
(In case it makes a difference, I know that $x, y$, and $z$ are each nonnegative integers and that they sum to some fixed $n$.)
On
I'm thinking that the answer is "no". Consider the simplified expression with only two variables \begin{equation} \log{\frac{(x+y)!}{x! y!}}, \end{equation} where $x+y=n$. This can be rewritten as \begin{equation} \log{\frac{n!}{x! (n-x)!}}. \end{equation} Plotting this function for $n=5$ gives
which is clearly nonlinear with respect to $x$.
Note that $\log x! = \log (1\cdot 2\cdot\ldots\cdot (x-1)\cdot x)= \sum_{i=1}^x \log i$. Thus we can do the following simplification, using the regular logarithm rules for handling multiplication and division.
$$\log{\frac{(x+y+z)!}{x! y! z!}} = \log (x+y+z)! - \log (x!y!z!) = $$ $$= \sum_{i=1}^{x+y+z}\log i - \sum_{i=1}^{x}\log i - \sum_{i=1}^{y}\log i -\sum_{i=1}^{z}\log i $$ It is not linear with respect to $x,y$ and $z$ though, but it is, in some sence, an easier expression.