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# 1. 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.

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# 5. 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.