a consequence of (\[eq.t\]) by setting $t=2$. By using generating functions, one can verify that the number of $2$-Dyck paths of length $2n$ equals the number of little Schröder paths of length $n$.

We now proceed to give a combinatorial proof of (\[eqt.t\]). To this end, we need to give a combinatorial interpretation of the entries in the matrix $M$ in (\[eq.t\]). We may define a composition of $k$-Dyck path $P$ as a sequence of $k$-Dyck paths $(P_1, P_2, \ldots, P_j)$ such that $P=P_1P_2\cdots P_j$, where $j$ is called the number of segments.

Let $a_{i,j}$ be the sum of weights of compositions of $(t^2-t)$-Dyck paths of length $2i$ with $j$ segments. Then $a_{i,j}$ satisfies the recurrence relation (\[eq.6\]).

The proof of the above lemma is similar to that of Lemma \[dotheom\]. Let us rewrite (\[eq.t\]) as follows $$\label{eqt.t}
\sum_{j\geq 1}m_{i,j}(t^j-(t-1)^j)=(t^2+t)^{i-1}.$$

In order to deal with $m_{i,j}(t^j-(t-1)^j)$ combinatorially, we introduce a coloring scheme on a composition of a $(t^2-t)$-Dyck path with $j$ segments. Suppose that we have $t$ colors $c_1, c_2,
\ldots, c_t$. If we use these $t$ colors to color the $j$ segments such that the first color $c_1$ must be used, then there are $t^j-(t-1)^j$ ways to accomplish such 