Geodesic
Returns and hills on generalized Dyck paths
Journal of integer sequences, Tome 19 (2016) no. 6.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: In 2009, Shapiro posed the following question: "What is the asymptotic proportion of Dyck paths having an even number of hills?" In this paper, we answer Shapiro's question, as well as a generalization of the question to ternary paths. We find that the probability that a randomly chosen ternary path has an even number of hills approaches 125/169 as the length of the path approaches infinity. Our strategy relies on properties of the Fine number sequence and extends certain relationships between the Catalan and Fine number generating functions.
Classification : 05A15, 05A16, 11B83
Keywords: Catalan number, ternary number, Dyck path, fine number, Motzkin number, Hill, return, lattice path statistics 
@article{JIS_2016__19_6_a1,
     author = {Cameron, Naiomi T. and Mcleod, Jillian E.},
     title = {Returns and hills on generalized {Dyck} paths},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
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     number = {6},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2016__19_6_a1/}
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Cameron, Naiomi T.; Mcleod, Jillian E. Returns and hills on generalized Dyck paths. Journal of integer sequences, Tome 19 (2016) no. 6. http://geodesic.mathdoc.fr/item/JIS_2016__19_6_a1/