Nondecreasing Dyck paths and q-Fibonacci numbers

In this paper, we introduce a subclass of the Dyck paths (Delest and Viennot, 1984) called nondecreasing Dyck paths which are enumerated by the Fibonacci numbers having odd indexes. We then use two different methods to enumerate these paths according to various parameters. By the first one, used in Barcucci et al. (1995, 1995), we determine an operator that allows us to construct each nondecreasing Dyck path p from another nondecreasing Dyck path p′ by performing a ‘local expansion’ on p′. Furthermore, every path p is obtained from only one path p′. Therefore, we obtain a nondecreasing Dyck path recursive description that allows us to deduce a functional equation verified by their generating function. By solving this functional equation, we obtain the nondecreasing Dyck pathsʹ generating function according to various parameters. This method does not allow us to investigate the area of nondecreasing Dyck paths. By the second method, we are able to give a recursive construction based on the definition of pyramid paths; from this definition we can immediatly deduce a functional equation verified by the nondecreasing Dyck pathsʹ generating function. In this case, we can also take the parameter area into account. By performing some calculations on this functional equation, we obtain the average area of all nondecreasing Dyck paths having a fixed length. This method allows us to introduce a q-generalization of Fibonacci numbers and interpret it from a combinatorial point of view.