  In 2003, Deutsch and Elizalde defined a family of bijective maps between the
set of Dyck paths to itself which is induced by some particular permutations.
In this paper, we extend the construction of the maps by allowing the
permutation to be arbitrary. We characterise the permutations which generate
the same map and find all permutations generating a bijection among Dyck paths.
Consequently, we give a new combinatorial interpretation of the quantity
$(2n-1)!!$ as well as some new statistics of Dyck paths which are
equidistributed to some known height statistics via our generalised maps.
