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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
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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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