The title says it all, there is a way to know how many partitions has a number?
With "a way" i mean if there is any formula or polynomials to provide the number of ways to "part" a number.
The question arose up after i answer the query "What are the partitions of 5?" I found seven, in fact is seven, but i would appreciate a double check
There is a function called the partition function which allows you to calculate exactly how many integer partitions there are of a given integer by calculating the coefficients of the $n^\textrm{th}$ term of the series $$(1+x+x^2+\cdots)(1+x^2+x^4 + \cdots)(1+x^3+x^6+\cdots)\cdots.$$ As you found, there are $7$ partitions of $5$ which are the following.
\begin{align} 5&=5 \\ 5&=4+1 \\ 5&=3+1+1 \\ 5&=2+1+1+1 \\ 5&=1+1+1+1+1 \\ 5&=2+3 \\ 5&=2+2+1 \end{align}
There isn't a known way to write these coefficients as a polynomial or anything, but we do know that it grows like $\frac{1}{4n\sqrt{3}}e^{\pi \sqrt{2n/3}}$ as $n \to \infty$. So we can figure out approximately how many integer partitions there are for large $n$.