Counting Ascents in Generalized Dyck Paths

Authors Benjamin Hackl , Clemens Heuberger , Helmut Prodinger

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Author Details

Benjamin Hackl
  • Alpen-Adria-Universität Klagenfurt, Austria
Clemens Heuberger
  • Alpen-Adria-Universität Klagenfurt, Austria
Helmut Prodinger
  • Department of Mathematical Sciences, Stellenbosch University, South Africa

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Benjamin Hackl, Clemens Heuberger, and Helmut Prodinger. Counting Ascents in Generalized Dyck Paths. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Non-negative Lukasiewicz paths are special two-dimensional lattice paths never passing below their starting altitude which have only one single special type of down step. They are well-known and -studied combinatorial objects, in particular due to their bijective relation to trees with given node degrees. We study the asymptotic behavior of the number of ascents (i.e., the number of maximal sequences of consecutive up steps) of given length for classical subfamilies of general non-negative Lukasiewicz paths: those with arbitrary ending altitude, those ending on their starting altitude, and a variation thereof. Our results include precise asymptotic expansions for the expected number of such ascents as well as for the corresponding variance.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Enumeration
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Mathematical software
  • Lattice path
  • Lukasiewicz path
  • ascent
  • asymptotic analysis
  • implicit function
  • singular inversion


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