Enumerating symmetric peaks in non-decreasing Dyck paths

Sergi Elizalde, Rigoberto Flórez, José Luis Ramírez

Abstract


Local maxima and minima of a Dyck path are called peaks and valleys, respectively. A Dyck path is non-decreasing if the heights (y-coordinates) of its valleys increase from left to right. A peak is symmetric if it is surrounded by two valleys (or endpoints of the path) at the same height. In this paper we give multivariate generating functions, recurrence relations, and closed formulas to count the number of symmetric and asymmetric peaks in non-decreasing Dyck paths. Finally, we use Riordan arrays to study weakly symmetric peaks, namely those for which the valley preceding the peak is at least as high as the valley following it.


Keywords


Non-decreasing Dyck path, symmetric peak, generating function, Riordan array, Fibonacci number

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DOI: https://doi.org/10.26493/1855-3974.2478.d1b

ISSN: 1855-3974

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