License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.AofA.2018.26
URN: urn:nbn:de:0030-drops-89191
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8919/
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Hackl, Benjamin ; Heuberger, Clemens ; Prodinger, Helmut

Counting Ascents in Generalized Dyck Paths

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LIPIcs-AofA-2018-26.pdf (0.4 MB)

Abstract

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.

BibTeX - Entry

@InProceedings{hackl_et_al:LIPIcs:2018:8919,
  author =	{Benjamin Hackl and Clemens Heuberger and Helmut Prodinger},
  title =	{{Counting Ascents in Generalized Dyck Paths}},
  booktitle =	{29th International Conference on Probabilistic,  Combinatorial and Asymptotic Methods for the Analysis of Algorithms  (AofA 2018)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-078-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{110},
  editor =	{James Allen Fill and Mark Daniel Ward},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8919},
  URN =		{urn:nbn:de:0030-drops-89191},
  doi =		{10.4230/LIPIcs.AofA.2018.26},
  annote =	{Keywords: Lattice path, Lukasiewicz path, ascent, asymptotic analysis, implicit function, singular inversion}
}

Keywords: Lattice path, Lukasiewicz path, ascent, asymptotic analysis, implicit function, singular inversion
Collection: 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)
Issue Date: 2018
Date of publication: 18.06.2018
Supplementary Material: Supplementary SageMath [SageMath Mathematics Software (Version 8.1), 2017] worksheets producing the results of this article can be found at https://benjamin-hackl.at/publications/lukasiewicz-ascents/.

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