More Models of Walks Avoiding a Quadrant

Authors Mireille Bousquet-Mélou, Michael Wallner

Thumbnail PDF


  • Filesize: 0.53 MB
  • 14 pages

Document Identifiers

Author Details

Mireille Bousquet-Mélou
  • CNRS, Université de Bordeaux, Laboratoire Bordelais de Recherche en Informatique, UMR 5800, 351 cours de la Libération, 33405 Talence Cedex, France
Michael Wallner
  • Université de Bordeaux, Laboratoire Bordelais de Recherche en Informatique, UMR 5800, 351 cours de la Libération, 33405 Talence Cedex, France
  • TU Wien, Institute for Discrete Mathematics and Geometry, Wiedner Hauptstraße 8 - 10, 1040 Wien, Austria


We thank our referees for their careful reading.

Cite AsGet BibTex

Mireille Bousquet-Mélou and Michael Wallner. More Models of Walks Avoiding a Quadrant. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We continue the enumeration of plane lattice paths avoiding the negative quadrant initiated by the first author in [Bousquet-Mélou, 2016]. We solve in detail a new case, the king walks, where all 8 nearest neighbour steps are allowed. As in the two cases solved in [Bousquet-Mélou, 2016], the associated generating function is proved to differ from a simple, explicit D-finite series (related to the enumeration of walks confined to the first quadrant) by an algebraic one. The principle of the approach is the same as in [Bousquet-Mélou, 2016], but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree. We also explain why we expect the observed algebraicity phenomenon to persist for 4 more models, for which the quadrant problem is solvable using the reflection principle.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Computations on polynomials
  • Enumerative combinatorics
  • lattice paths
  • non-convex cones
  • algebraic series
  • D-finite series


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. M. Bousquet-Mélou. Square lattice walks avoiding a quadrant. J. Combin. Theory Ser. A, 144:37-79, 2016. URL:
  2. M. Bousquet-Mélou and A. Jehanne. Polynomial equations with one catalytic variable, algebraic series and map enumeration. J. Combin. Theory Ser. B, 96:623-672, 2006. URL:
  3. M. Bousquet-Mélou and M. Mishna. Walks with small steps in the quarter plane. In Algorithmic probability and combinatorics, volume 520 of Contemp. Math., pages 1-39. Amer. Math. Soc., Providence, RI, 2010. URL:
  4. D. Denisov and V. Wachtel. Random walks in cones. Ann. Probab., 43(3):992-1044, 2015. URL:
  5. G. Fayolle, R. Iasnogorodski, and V. Malyshev. Random walks in the quarter-plane: Algebraic methods, boundary value problems and applications, volume 40 of Applications of Mathematics. Springer-Verlag, Berlin, 1999. Google Scholar
  6. I. Gessel. A factorization for formal Laurent series and lattice path enumeration. J. Combin. Theory Ser. A, 28(3):321-337, 1980. URL:
  7. I. M. Gessel and D. Zeilberger. Random walk in a Weyl chamber. Proc. Amer. Math. Soc., 115(1):27-31, 1992. URL:
  8. L. Lipshitz. The diagonal of a D-finite power series is D-finite. J. Algebra, 113(2):373-378, 1988. URL:
  9. L. Lipshitz. D-finite power series. J. Algebra, 122:353-373, 1989. URL:
  10. M. Mishna and A. Rechnitzer. Two non-holonomic lattice walks in the quarter plane. Theoret. Comput. Sci., 410(38-40):3616-3630, 2009. URL:
  11. S. Mustapha. Non-D-finite walks in a three-quadrant cone. Ann. Comb., 23(1):143-158, 2019. URL:
  12. K. Raschel. Counting walks in a quadrant: a unified approach via boundary value problems. J. Eur. Math. Soc. (JEMS), 14(3):749-777, 2012. URL:
  13. K. Raschel and A. Trotignon. On walks avoiding a quadrant. Electron. J. Combin., 26(3):Paper 3.31, 34, 2019. URL:
  14. B. Salvy and P. Zimmermann. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, 20(2):163-177, 1994. URL: