# Bose Einstein Coefficients

A collection of $n$ objects that can have one of $k$ different classes are selected. The classes have equal probability. How many ways are there to select the objects? 

Alternatively: how many ways are there to partition $n$ objects into $k$ buckets? 

If $n_i$ is the number of objects in bucket $i$, then $\sum_i n_i = n$.


## Stars and Bars

This is a neat way to approach counting problems.

Dividing objects into $k$ classes requires $k-1$ separators. The division can be visualized of a sequence of $n+k-1$ objects and dividers.

Consider 10 objects divided over 4 classes. 

That makes 10 objects and 3 dividers that are allocated to $n+k-1$ slots:

``[ , , , , , , , , , , , , ]``

For example:

``[|,* ,* ,* ,|,* ,| ,* ,* ,* ,* ,* ,* ]``

Corresponds to an allocation of $\{0,3,1,6\}$ across the 4 classes. 

The amount of ways to distribute $n$ objects over $n+k-1$ is:

\begin{equation}
\left(\begin{array}{c}n+k-1\\n\end{array}\right)
\end{equation}

Which is the result.

The Bose Einstein Coefficient is also known as the multichoose coefficient. It counts the ways that $n$ indistinct objects can be assigned to $k$ classes. 

# Rising Factorials

What if the the objects that are distributed are distinct, so that they can have an internal ordering? For example, consider the case of $n$ books being distributed over $k$ library shelves.

In that case, there are 

\begin{equation}
\left(\begin{array}{c}n+k-1\\n\end{array}\right)
\end{equation}

Ways of dividing the books up between the shelves, and 

\begin{equation}
n! 
\end{equation}

ways for the books to be ordered internally. 


For example:

``[|,1 ,3 ,6 ,|,4 ,| ,5 ,7 ,9 ,8 ,11 ,10 ]``

The amount of combinations is:

\begin{equation}
n! {n + k - 1 \choose n} = \frac{(n+k-1)!}{(k-1)!} = (n+k-1)^\underline{n} = k^\overline{n}
\end{equation}



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Combinatorics - Stars and Bars, Bose Einstein Coefficients.ipynb
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Combinatorics - Stars and Bars, Bose Einstein Coefficients.ipynb
jpbm/probabilism
a2f5c1595aed616236b2b889195604f365175899
MIT
Combinatorics - Stars and Bars, Bose Einstein Coefficients.ipynb
jpbm/probabilism
a2f5c1595aed616236b2b889195604f365175899
MIT
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