  We prove new bijections between different variants of Dyck paths and integer
compositions, which give combinatorial explanations of their simple counting
formula $4^{n-1}$. These give relations between different statistics, such as
the number of crossings of the $x$-axis in classes of Dyck bridges or the
distribution of peaks in classes of Dyck paths, and furthermore relate them
with $k$- and $g$-compositions. These allow us to find and prove congruence
results for Dyck paths and parity results for compositions. Our investigation
uncovers unexpected connections to mock theta functions, Hardinian arrays,
little Schr\"oder paths, Fibonacci numbers, and irreducible pairs of
compositions, offering new insights into the structures of paths, partitions
and compositions.
