Restricting Dyck Paths and 312-Avoiding Permutations

We interpret Dyck paths of height at most h and without valleys at height h − 1 combinatorially, by means of 312-avoiding permutations with some restrictions on their left-to-right maxima. We obtain our results by analyzing a restriction of a well-known bijection between the sets of Dyck paths and 312-avoiding permutations. We also provide a recursive formula enumerating these two structures by using the ECO method and the theory of production matrices. As a further result we obtain a family of combinatorial identities involving Catalan numbers.