# number of partitioning identical objects

• Jun 3rd 2013, 03:34 PM
Yuuki
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?
• Jun 3rd 2013, 04:11 PM
Plato
Re: number of partitioning identical objects
Quote:

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.