The Kernel Method for Lattice Paths Below a Line of Rational Slope

  • Cyril Banderier
  • Michael WallnerEmail author
Part of the Developments in Mathematics book series (DEVM, volume 58)


We analyze some enumerative and asymptotic properties of lattice paths below a line of rational slope. We illustrate our approach with Dyck paths under a line of slope 2 / 5. This answers Knuth’s problem #4 from his “Flajolet lecture” during the conference “Analysis of Algorithms” (AofA’2014) in Paris in June 2014. Our approach extends the work of Banderier and Flajolet for asymptotics and enumeration of directed lattice paths to the case of generating functions involving several dominant singularities and has applications to a full class of problems involving some “periodicities.” A key ingredient in the proof is the generalization of an old trick by Knuth himself (for enumerating permutations sortable by a stack), promoted by Flajolet and others as the “kernel method.” All the corresponding generating functions are algebraic, and they offer some new combinatorial identities, which can also be tackled in the A = B spirit of Wilf–Zeilberger–Petkovšek. We show how to obtain similar results for any rational slope. An interesting case is, e.g.,  Dyck paths below the slope 2 / 3 (this corresponds to the so-called Duchon’s club model), for which we solve a conjecture related to the asymptotics of the area below such lattice paths. Our work also gives access to lattice paths below an irrational slope (e.g., Dyck paths below \(y=x/\sqrt{2}\)), a problem that we study in a companion article.


Lattice paths Generating function Analytic combinatorics Singularity analysis Kernel method Generalized Dyck paths Algebraic function Rational Catalan combinatorics Periodic support Bizley formula Grossman formula 

2010 Mathematics Subject Classification

Primary 05A15 Secondary 05A16 68W40 



This work is the result of a collaboration founded by the SFB project F50 “Algorithmic and Enumerative Combinatorics” and the Franco-Austrian PHC “Amadeus.” Michael Wallner is supported by the Austrian Science Fund (FWF) grant SFB F50-03 and by ÖAD, grant F04/2012. A preliminary version of this work [14] was presented at the conference ANALCO’15 (San Diego, January 2015) and at the 8th International Conference on Lattice Path Combinatorics and Applications (Pomona, August 2015). Last but not least, we thank Don Knuth, Ernst Schulte-Geers and Manuel Kauers for exchanging references on this problem, and the referee for the detailed feedback!


  1. 1.
    Aebly, J.: Démonstration du problème du scrutin par des considérations géométriques. Enseign. Math. 23, 185–186 (1923)zbMATHGoogle Scholar
  2. 2.
    André, D.: Solution directe du problème résolu par M. Bertrand. C. R. Acad. Sci. Paris 105, 436–437 (1887)zbMATHGoogle Scholar
  3. 3.
    Armstrong, D., Rhoades, B., Williams, N.: Rational Catalan combinatorics: the associahedron. Discret. Math. Theor. Comput. Sci. Proc. AS 933–944 (2013)Google Scholar
  4. 4.
    Bacher, A., Beaton, N.: Weakly prudent self-avoiding bridges. Discret. Math. Theor. Comput. Sci. Proc. AT 827–838 (2014)Google Scholar
  5. 5.
    Banderier, C.: Combinatoire analytique : application aux marches aléatoires. Mémoire de DEA, INRIA Rocquencourt/Univ. Paris 6 (1998)Google Scholar
  6. 6.
    Banderier, C., Bousquet-Mélou, M., Denise, A., Flajolet, P., Gardy, D., Gouyou-Beauchamps, D.: Generating functions for generating trees. Discret. Math. 246(1–3), 29–55 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Banderier, C., Drmota, M.: Formulae and asymptotics for coefficients of algebraic functions. Combin. Probab. Comput. 24(1), 1–53 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Banderier, C., Flajolet, P.: Basic analytic combinatorics of directed lattice paths. Theor. Comput. Sci. 281(1–2), 37–80 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Banderier, C., Flajolet, P., Schaeffer, G., Soria, M.: Random maps, coalescing saddles, singularity analysis, and Airy phenomena. Random Struct. Algorithms 19(3–4), 194–246 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Banderier, C., Gittenberger, B.: Analytic combinatorics of lattice paths: enumeration and asymptotics for the area. Discret. Math. Theor. Comput. Sci. Proc. AG 345–355 (2006)Google Scholar
  11. 11.
    Banderier, C., Krattenthaler, C., Krinik, A., Kruchinin, D., Kruchinin, V., Nguyen, D., Wallner, M.: Explicit formulas for enumeration of lattice paths: basketball and the kernel method. This volumeGoogle Scholar
  12. 12.
    Banderier, C., Nicodème, P.: Bounded discrete walks. Discret. Math. Theor. Comput. Sci. Proc. AM 35–48 (2010)Google Scholar
  13. 13.
    Banderier, C., Schwer, S.: Why Delannoy numbers? J. Stat. Plan. Inference 135(1), 40–54 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Banderier, C., Wallner, M.: Lattice paths of slope \(2/5\). In: Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pp. 105–113 (2015)Google Scholar
  15. 15.
    Banderier, C., Wallner, M.: Lattice paths below a line of irrational slope. in preparation (2016)Google Scholar
  16. 16.
    Bertrand, J.: Solution d’un problème. C. R. Acad. Sci. Paris 105, 369 (1887)zbMATHGoogle Scholar
  17. 17.
    Bizley, M.T.L.: Derivation of a new formula for the number of minimal lattice paths from \((0,0)\) to \((km, kn)\) having just \(t\) contacts with the line \(my=nx\) and having no points above this line; and a proof of Grossman’s formula for the number of paths which may touch but do not rise above this line. J. Inst. Actuar. 80, 55–62 (1954)Google Scholar
  18. 18.
    Bostan, A., Bousquet-Mélou, M., Kauers, M., Melczer, S.: On 3-dimensional lattice walks confined to the positive octant. Ann. Combin. 20, 661–704 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bousquet-Mélou, M.: Discrete excursions. Sém. Lothar. Combin. 57, 23 (2008). Article B57dGoogle Scholar
  20. 20.
    Bousquet-Mélou, M., Fusy, É., Préville-Ratelle, L.-F.: The number of intervals in the \(m\)-Tamari lattices. Electron. J. Combin. 18(2), 26 (2011). Article #R31Google Scholar
  21. 21.
    Bousquet-Mélou, M., Jehanne, A.: Polynomial equations with one catalytic variable, algebraic series and map enumeration. J. Combin. Theory Ser. B 96(5), 623–672 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Bousquet-Mélou, M., Mishna, M.: Walks with small steps in the quarter plane. Algorithmic Probab. Combinatorics. Contemp. Math. 520, 1–39 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Bousquet-Mélou, M., Petkovšek, M.: Linear recurrences with constant coefficients: the multivariate case. Discret. Math. 225(1–3), 51–75 (2000)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Brown, W.G.: On the existence of square roots in certain rings of power series. Math. Ann. 158, 82–89 (1965)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Chung, F., Graham, R., Morrison, J., Odlyzko, A.: Pebbling a chessboard. Am. Math. Mon. 102(2), 113–123 (1995)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Denisov, D., Wachtel, V.: Random walks in cones. Ann. Probab. 43(3), 992–1044 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Duchi, E.: On some classes of prudent walks. In: Proceedings of FPSAC’2005, Taormina, Italy (2015)Google Scholar
  28. 28.
    Duchon, P.: On the enumeration and generation of generalized Dyck words. Discret. Math. 225(1–3), 121–135 (2000)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Dvoretzky, A., Motzkin, T.: The asymptotic density of certain sets of real numbers. Duke Math. J. 14, 315–321 (1947)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Eynard, B.: Counting Surfaces. Progress in mathematical physics, vol. 70. Birkhäuser/Springer [Cham] (2016)Google Scholar
  31. 31.
    Fayolle, G., Iasnogorodski, R., Malyshev, V.: Random Walks in the Quarter-plane. Applications of mathematics, vol. 40. Springer, Berlin (1999)Google Scholar
  32. 32.
    Flajolet, P.: Combinatorial aspects of continued fractions. Discret. Math. 32(2), 125–161 (1980)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)Google Scholar
  34. 34.
    Gaudin, M.: The Bethe Wavefunction. Cambridge University Press, New York (2014)Google Scholar
  35. 35.
    Gessel, I.M.: A factorization for formal Laurent series and lattice path enumeration. J. Combin. Theory Ser. A 28(3), 321–337 (1980)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Hardy, G.H., Ramanujan, S.: Asymptotic formulae for the distribution of integers of various types. Proc. Lond. Math. Soc. 16(2), 112–132 (1917). Collected papers of Srinivasa Ramanujan, pp. 245–261. AMS Chelsea Publication, Providence (2000)Google Scholar
  37. 37.
    Jain, J.L., Mohanty, S.G., Böhm, W.: A Course on Queueing Models. Statistics: textbooks and monographs. Chapman and Hall/CRC, FL (2007)Google Scholar
  38. 38.
    Janse van Rensburg, E.J.: Square lattice directed paths adsorbing on the line \(y = qx\). J. Stat. Mech.: Theory Exp. 2005(09), P09010 (2005)Google Scholar
  39. 39.
    Janse van Rensburg, E.J., Prellberg, T., Rechnitzer, A.: Partially directed paths in a wedge. J. Combin. Theory Ser. A 115(4), 623–650 (2008)Google Scholar
  40. 40.
    Janse van Rensburg, E.J., Rechnitzer, A.: Adsorbing and collapsing directed animals. J. Stat. Phys. 105(1–2), 49–91 (2001)Google Scholar
  41. 41.
    Kauers, M., Paule, P.: The Concrete Tetrahedron. Texts and monographs in symbolic computation. Springer, Berlin (2011)Google Scholar
  42. 42.
    Kempner, A.J.: A theorem on lattice-points. Ann. Math. 19(2), 127–136 (1917)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Knuth, D.E.: The Art of Computer Programming, vol. 1: Fundamental Algorithms. Addison–Wesley (1968)Google Scholar
  44. 44.
    Knuth, D.E.: 20th Annual Christmas Tree Lecture: (3/2)-ary Trees. Stanford University (2014).
  45. 45.
    Krattenthaler, C.: Lattice Path Enumeration. In: Bóna, M. (ed.) Handbook of Enumerative Combinatorics. Discrete Mathematics and its Applications, pp. 589–678. CRC Press (2015)Google Scholar
  46. 46.
    Kurkova, I., Raschel, K.: On the functions counting walks with small steps in the quarter plane. Publ. Math. Inst. Hautes Études Sci. 116, 69–114 (2012)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Labelle, J., Yeh, Y.-N.: Generalized Dyck paths. Discret. Math. 82(1), 1–6 (1990)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Lagrange, J.-L.: Nouvelle méthode pour résoudre les équations littérales par le moyen des séries. Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Berlin 24, 251–326 (1770). Reprinted in Œuvres de Lagrange, tome 2, pp. 655–726. Gauthier-Villars, Paris (1868)Google Scholar
  49. 49.
    Mansour, T., Shattuck, M.: Pattern avoiding partitions, sequence A054391 and the kernel method. Appl. Appl. Math. 6(12), 397–411 (2011)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Merlini, D., Sprugnoli, R., Verri, M.C.: The area determined by underdiagonal lattice paths. In: Proceedings of CAAP’96. Lecture Notes in Computer Science, vol. 1059, pp. 59–71 (1996)Google Scholar
  51. 51.
    Mirimanoff, D.: À propos de l’interprétation géométrique du problème du scrutin. Enseign. Math. 23, 187–189 (1923)zbMATHGoogle Scholar
  52. 52.
    Mohanty, S.G.: Lattice Path Counting and Applications. Academic Press, New York (1979)Google Scholar
  53. 53.
    Nakamigawa, T., Tokushige, N.: Counting lattice paths via a new cycle lemma. SIAM J. Discret. Math. 26(2), 745–754 (2012)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Narayana, T.V.: Lattice Path Combinatorics with Statistical Applications. Mathematical Expositions, vol. 23. University of Toronto Press (1979)Google Scholar
  55. 55.
    Petkovšek, M.: The irrational chess knight. In: Proceedings of FPSAC’98, pp. 513–522 (1998)Google Scholar
  56. 56.
    Petkovšek, M., Wilf, H.S., Zeilberger, D.: A \(=\) B. AK Peters (1996)Google Scholar
  57. 57.
    Prodinger, H.: The kernel method: a collection of examples. Sém. Lothar. Combin. 50, 19 (2003/2004). Article B50fGoogle Scholar
  58. 58.
    Salvy, B., Zimmermann, P.: Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Trans. Math. Softw. 20(2), 163–177 (1994)CrossRefGoogle Scholar
  59. 59.
    Sato, M.: Generating functions for the number of lattice paths between two parallel lines with a rational incline. Math. Japon. 34(1), 123–137 (1989)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Schulte-Geers, E., Stadje, W.: Maximal percentages in Pólya’s urn. J. Appl. Probab. 52(1), 180–190 (2015)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Schwerdtfeger, U.: Linear functional equations with a catalytic variable and area limit laws for lattice paths and polygons. Eur. J. Combin. 36, 608–640 (2014)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999)Google Scholar

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Authors and Affiliations

  1. 1.Laboratoire d’Informatique de Paris-NordCNRS/Université Paris 13VilletaneuseFrance
  2. 2.Institute of Discrete Mathematics and GeometryTU WienViennaAustria
  3. 3.LaBRIUniversité de BordeauxBordeauxFrance

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