The sum of the skew hook positions in a Dyck path.
A skew hook is an occurrence of a down step followed by two up steps or of an up step followed by a down step.
Write $U_i$ for the $i$-th up step and $D_j$ for the $j$-th down step in the Dyck path. Then the skew hook set is the set $$H = \{j: U_{i−1} U_i D_j \text{ is a skew hook}\} \cup \{i: D_{i−1} D_i U_j\text{ is a skew hook}\}.$$
This statistic is the sum of all elements in $H$.

Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
The Delest-Viennot bijection $\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path $(\gamma^{(-1)}\circ\beta)(D)$.

The bounce path determined by an integer composition.

The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.