number of partitioning identical objects

I've studied about a third of my combinatorics textbook, and I'm beginning to lose track of the many "numbers" and methods of computations.

I know that the number of ways to distribute **n distinct objects into k identical classes** is given by the stirling numbers of the second kind.

Was there an explicit formula for partitioning **n identical objects into k identical classes**?

Re: number of partitioning identical objects

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Originally Posted by

**Yuuki** Was there an explicit formula for partitioning **n identical objects into k identical classes**?

These are known as **integer partitions**.

For example:

$\displaystyle \begin{array}{*{20}{c}} 5&{4 + 1}&{3 + 1 + 1}&{2 + 1 + 1 + 1} \\ {}&{3 + 2}&{2 + 2 + 1}&{1 + 1 + 1 + 1 + 1} \end{array}$

Those are the partitions of five identical objects into five or fewer identical classes.

Now you can see that if a class can be empty or not matters. That is why the phrase "or fewer identical classes" is added.

The rule for this is defined by a recursive function.

One good source here is __MATHEMATICS OF CHOICE__ by Ivan Niven.