Dyck paths having height at most h and without valleys at height hβ1 are
combinatorially interpreted by means of 312-avoding permutations with some
restrictions on their \emph{left-to-right maxima}. The results are obtained by
analyzing a restriction of a well-known bijection between the sets of Dyck
paths and 312-avoding permutations. We also provide a recursive formula
enumerating these two structures using ECO method and the theory of production
matrices. As a further result we obtain a family of combinatorial identities
involving Catalan numbers