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Math Help - number of partitioning identical objects

  1. #1
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    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?
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  2. #2
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    Re: number of partitioning identical objects

    Quote Originally Posted by Yuuki View Post
    Was there an explicit formula for partitioning n identical objects into k identical classes?
    These are known as integer partitions.
    For example:
    \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.
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