Abstract
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 thek-generalized Fibonacci numbers, the powers of 2, the Pell numbers, thek-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 afull historyrecurrence relation. The whole study can be accomplished also by involving particular sets of stringsviaa simple encoding of Dyck paths.
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