I believe this to be a question in Combinatorics. To state more formally, given a labeled cycle graph with $n$ vertices $C_{n}$ and parameters $a_{1}, ..., a_{k}$, such that $a_{1}+...+a_{k}=n$. How many ways can we decompose $C_{n}$ into consecutive paths of lengths $a_{1}, ..., a_{k}$?
In my thought, it would be relevant to the values of $a_{1}, ..., a_{k}$. Define $\sigma$ to be the order of $a_{1}, ..., a_{k}$, which means $\sigma$ is the minimal positive integer, such that $a_{1}=a_{1+\sigma(mod k)}$, ..., $a_{k}=a_{k+\sigma(mod k)}$. Then the number of ways would be $\frac{n\sigma}{k}$.
I wonder if there is an explicit formula for this. I didn't find relevant questions and wonder if that's because my improper way of statement.