<> 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 $\sum_{1 \leq i \leq k} \alpha_i = n$. <> - 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 ====================== **tba** References ========== - [Wikipedia](http://en.wikipedia.org/wiki/Composition_%28number_theory%29) Sage examples ============= {{{#!sagecell for n in [2,3,4,5]: print Compositions(n).cardinality() for c in Compositions(3): print c }}} 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.