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Definition & Example
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- An **integer composition** $\alpha$ of $n \in \mathbb{N}_+$ is a sequence $\alpha = (\alpha_1,\ldots,\alpha_k)$ such that $\alpha_i \in \mathbb{N}_{+}$ and $\sum_{1 \leq i \leq k} \alpha_i = n$.
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- There are $2^{n-1}$ integer compositions of $n$,n, see [A000079](https://oeis.org/A000079), and $\binom{n-1}{k}$ integer compositions of $n$ into $k$ parts, see [A007318](https://oeis.org/A007318).
Additional information
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**tba**
References
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- [Wikipedia](http://en.wikipedia.org/wiki/Composition_%28number_theory%29)
Sage examples
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{{{#!sagecell
for n in [2,3,4,5]:
print Compositions(n).cardinality()
for c in Compositions(3):
print c
}}}
Technical information for database usage
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- An integer composition is uniquely **represented as a list of its parts**.
- Integer compositions are **graded by their sum**.
- The database contains all integer compositions of size at most 10.