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- W3133600612 abstract "The notion of symmetric and asymmetric peaks in Dyck paths was introduced by Flórez and Rodríguez, who counted the total number of such peaks over all Dyck paths of a given length. In this paper we generalize their results by giving multivariate generating functions that keep track of the number of symmetric peaks and the number of asymmetric peaks, as well as the widths of these peaks. We recover a formula of Denise and Simion as a special case of our results. We also consider the analogous but more intricate notion of symmetric valleys. We find a continued fraction expression for the generating function of Dyck paths with respect to the number of symmetric valleys and the sum of their widths, which provides an unexpected connection between symmetric valleys and statistics on ordered rooted trees. Finally, we enumerate Dyck paths whose peak or valley heights satisfy certain monotonicity and unimodality conditions, using a common framework to recover some known results, and relating our questions to the enumeration of certain classes of column-convex polyominoes." @default.
- W3133600612 created "2021-03-15" @default.
- W3133600612 creator A5091007333 @default.
- W3133600612 date "2021-06-01" @default.
- W3133600612 modified "2023-09-23" @default.
- W3133600612 title "Symmetric peaks and symmetric valleys in Dyck paths" @default.
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- W3133600612 doi "https://doi.org/10.1016/j.disc.2021.112364" @default.
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