I'm looking for an alternative to the following (possibly standard) proof for the number of weak compositions:

The number of $k$-compositions of $n+k$ corresponds to a weak one by subtracting 1 from each "bin". Thus we have $\binom{n+k-1}{k-1}$.

While I kind of like this proof and always felt it made sense lately I've been left wanting something a little more convincing. Does anyone have a different proof of this result?


The argument that you outlined should be completely convincing: it displays a bijection between the set of weak $k$-compositions of $n$ and the number of $k$-compositions of $n+k$.

It’s certainly possible to give another argument, however. Line up $k-1$ dividers, and distribute your $n$ objects into the $k-1$ spaces between adjacent dividers, before the first divider, and after the last one. Each possible distribution of objects gives you a weak $k$-composition of $n$, and vice versa. Each of these distributions can be regarded as a string of $n$ objects and $k-1$ dividers, and the corresponding weak composition is completely determined by the location of the dividers in the string. There are $\binom{n+k-1}{k-1}$ ways to place the $k-1$ dividers in the string, so there are $\binom{n+k-1}{k-1}$ weak compositions of $n$.

This is one of the two arguments commonly known as stars-and-bars arguments.


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