Integer Compositions

# 1. Definition

An integer composition $\alpha$ of $n \in \mathbf{N}$ is a sequence $\alpha_1,\ldots,\alpha_m$ such that $\alpha_m > 0$ and $\sum_{1 \leq k \leq m} \alpha_k = n$.

# 2. Notation

By $\mathbf{C}_n$, we denote the collection of all integer compositions of $n$. We usually write $\alpha \in \mathbf{C}_n$ as $\alpha = (\alpha_1,\alpha_2, \ldots)$. Integer compositions of $n$ are in bijection with the subsets of $\{1,2,\ldots, n-1\}$. The set corresponding to the composition $\alpha$ will be denoted $D(\alpha)$ and is equal to the set $\{ \alpha_1, \alpha_1 +\alpha_2, \ldots, \alpha_1+\alpha_2+ \cdots + \alpha_{m-1} \}$. The composition corresponding to a set $D = \{ d_1, d_2, \ldots, d_{m-1} \}$ will be denoted $\alpha(D)$ and is equal to the composition $(d_1, d_2-d_1, d_3-d_2,\ldots, d_{m-1} - d_{m-2}, n-d_{m-1})$.

There are three obvious involutions on compositions corresponding to the reverse of the list, complementation of the set, and the composition of these two operations. They will be denoted as $\alpha^r$, $\alpha^c$ and $\alpha'$ where they are defined as $$\alpha^r = (\alpha_m, \alpha_{m-1}, \ldots, \alpha_1) = \alpha(\{ n-i : i \in D(\alpha) \})$$ $$\alpha^c = \alpha(\{1,2,\ldots, n-1\} - D(\alpha)),$$ $$\hbox{ and }\alpha' = (\alpha^r)^c = (\alpha^c)^r.$$

There are several ways of representing a composition with a picture. Sometimes they are thought of as a skew partition $\lambda/{\overline \lambda}$ where $$\lambda = (\alpha_1+\alpha_2+\cdots+\alpha_{m}-m+1, \alpha_1+\alpha_2+\cdots+\alpha_{m-1}-m+2, \ldots, \alpha_1+\alpha_2-1, \alpha_1)\hbox{ and }{\overline \lambda} = (\lambda_2-1,\lambda_3-1,\ldots,\lambda_m-1).$$ Also present example of **bar graph** and/or **skyline diagram**

# 3. Examples

$\mathbf{C}_1 = \{ [1]\}$

$\mathbf{C}_2 = \{ [2],[1,1]\}$

$\mathbf{C}_3 = \{ [3],[2,1],[1,2],[1,1,1]\}$

$\mathbf{C}_4 = \{ [4],[3,1],[1,3],[2,2], [2,1,1], [1,2,1], [1,1,2], [1,1,1,1]\}$

# 4. Properties

$|\mathbf{C}_n| = 2^{n-1}$.

# 5. Statistics

We have the following **23 statistics** in the database:

# 6. Maps

We have the following **5 maps** in the database:

# 7. References

# 8. Sage examples