I was wondering if there is a formula for calculating the number of unique arrangements of a discrete data set of positive integers of size $n$, that result in a given mean of lets say $x$. I stress positive integers as I imagine if we allowed for negative values the number of combinations would be infinite, and if we allowed for non-integer values the same situation would arise.
I have little experience in notation of multivariate functions, but I believe in mathematical terms this is what I am looking for: $$c = f(x,n)$$ Where, $x$ is the given mean, $n$ is the number of data items, and $c$ is the number of unique arrangements of the data set size $n$ that have a mean of $x$.
If you specify the mean and the nunmber of integers, you're also specifying the sum, so you're basically just asking for the number of ways to partition a given sum $nx$ into $n$ positive integers. Depending on whether you want to distinguish the partitions by order, this is either the partition function or a standard stars and bars count of $\binom{nx-1}{n-1}$.