Sequences from Fibonacci to Catalan: A combinatorial interpretation via Dyck paths

We use Dyck paths having some restrictions in order to give a combinatorial interpretation for some famous number sequences. Starting from the Fibonacci numbers we show how the k-generalized Fibonacci numbers, the powers of 2, the Pell numbers, the k-generalized Pell numbers and the even-indexed Fibonacci numbers can be obtained by means of constraints on the number of consecutive valleys (at a given height) of the Dyck paths. By acting on the maximum height of the paths we get a succession of number sequences whose limit is the sequence of Catalan numbers. For these numbers we obtain a family of interesting relations including a full history recurrence relation. The whole study can be accomplished also by involving particular sets of strings via a simple encoding of Dyck paths.