Number of ways to distribute identical balls into identical bins

Question

I have a question which I expected to be quite famous and common, yet I haven't found much...

How many ways are there to distribute $k$ identical balls into $n$ identical bins?

For example $(k, n) = (6, 4)$:

\begin{align} 6 &= (6, 0, 0, 0) \\ &= (5, 1, 0, 0) \\ &= (4, 2, 0, 0) \\ &= (4, 1, 1, 0) \\ &= (3, 3, 0, 0) \\ &= (3, 2, 1, 0) \\ &= (3, 1, 1, 1) \\ &= (2, 2, 2, 0) \\ &= (2, 2, 1, 1) \end{align}

So there are $9$ ways how to distribute the balls. I've went through this site but I've found just these questions:

In how many ways can five letters be posted in 4 boxes?
Distributing identical objects to identical boxes

For example I learned that the number of ways to distribute labeled balls into identical bins has something in common with the Stirling numbers of the second kind. I expected my question to be somehow famous and with deep underlying math as well... Do you know anything about it?

Answer 1

To make my question answered:

Thank to @Shaktal and @AndréNicolas I learned that distributing identical objects into identical bins is closely connected with a part of number theory called Integer partitions. Here's an article that provides a nice amount of details, if anyone in the future is tackling the same topic:

http://en.wikipedia.org/wiki/Partition_(number_theory)