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# Definition & Example

- 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 $\sum_{1 \leq i \leq k} \alpha_i = n$.

the 16 Integer compositions of size 5 | |||||||||||||||

[1,1,1,1,1] |
[1,1,1,2] |
[1,1,2,1] |
[1,1,3] |
[1,2,1,1] |
[1,2,2] |
[1,3,1] |
[1,4] |
||||||||

[2,1,1,1] |
[2,1,2] |
[2,2,1] |
[2,3] |
[3,1,1] |
[3,2] |
[4,1] |
[5] |

- There are $2^{n-1}$ integer compositions of $n$,n, see A000079, and $\binom{n-1}{k}$ integer compositions of $n$ into $k$ parts, see A007318.

# Additional information

**tba**

# References

# Sage examples

# Technical information for database usage

- 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.

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