ending point is always a return point. Formally speaking, a [*composition*]{} of a Dyck path $P$ is sequence of Dyck path $(P_1,
P_2, \ldots, P_j)$ such that $P=P_1 P_2 \cdots P_j$, where $P_1,
P_2, \ldots, P_j$ are Dyck paths. For a composition $(P_1, P_2,
\ldots, P_j)$ of a Dyck path $P$, its length is meant to be the length of $P$ and $j$ is called the number of segments. We may choose certain return points to cut off the Dyck paths into a composition. We use the convention that the ending point is always a cut point. Clearly, a Dyck path with one segment is an ordinary Dyck path.

\[dotheom\] For $j\geq 2$, we have the following recurrence relation $$\label{eq.1}
 d_{i,j}=d_{i-1,j-1} +
2d_{i-1,j} + d_{i-1,j+1}.$$

Let $(P_1, P_2, \ldots, P_{j})$ be composition of a Dyck path $P$ of length $2i$. Consider the following cases for $P_1$. Case 1: $P_1=UD$. Then we get a composition length $2(i-1)$ with $j-1$ segments: $(P_2, \ldots, P_j)$. Case 2: $P_1=QUD$ and $Q$ is not empty. Then we get a composition $(Q, P_2, \ldots, P_j)$ of length $2(i-1)$ and $j$ segments. Case 3: $P_1=U Q D$, $Q$ is not empty. We get a composition $(Q,  P_2, 