, Volume 75, Issue 4, pp 782–811 | Cite as

Asymptotic Lattice Path Enumeration Using Diagonals

  • Stephen Melczer
  • Marni MishnaEmail author


We consider d-dimensional lattice path models restricted to the first orthant whose defining step sets exhibit reflective symmetry across every axis. Given such a model, we provide explicit asymptotic enumerative formulas for the number of walks of a fixed length: the exponential growth is given by the number of distinct steps a model can take, while the sub-exponential growth depends only on the dimension of the underlying lattice and the number of steps moving forward in each coordinate. The generating function of each model is first expressed as the diagonal of a multivariate rational function, then asymptotic expressions are derived by analyzing the singular variety of this rational function. Additionally, we show how to compute subdominant growth, reflect on the difference between rational diagonals and differential equations as data structures for D-finite functions, and show how to determine first order asymptotics for the subset of walks that start and end at the origin.


Lattice path enumeration D-finite Diagonal Analytic combinatorics in several variables Weyl chambers 



The authors would like to thank Manuel Kauers for the construction in Proposition 2.6, and illuminating discussions on diagonals of generating functions, and the anonymous referees and editors of both this work and its previous extended abstract for their comments and suggestions. We are also grateful to Mireille Bousquet-Mélou for pointing out some key references and provoking important clarifications.


  1. 1.
    André, D.: Solution directe du problème résolu par M. Bertrand. C. R. Acad. Sci. Paris 105, 436–437 (1887)Google Scholar
  2. 2.
    Aparicio-Monforte, A., Kauers, M.: Formal Laurent series in several variables. Expos. Math. 31(4), 350–367 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bostan, A., Bousquet-Mélou, M., Kauers, M., Melczer, S.: On lattice walks confined to the positive octant. Accepted to the Ann. Comb. November (2014).
  4. 4.
    Bostan, A., Kauers, M.: Automatic classification of restricted lattice walks. In: Proceedings of FPSAC 2009, Discrete Mathematics and Theoretical Computer Science Proceedings, AK, pp. 201–215 (2009)Google Scholar
  5. 5.
    Bostan, A., Kurkova, I., Raschel, K.: A human proof of Gessel’s lattice path conjecture.
  6. 6.
    Bostan, A., Lairez, P., Salvy, B.: Creative telescoping for rational functions using the Griffiths–Dwork method. In: Proceedings of the international symposium on symbolic and algebraic computation (ISSAC), New York, NY, USA. ACM, pp. 93–100 (2013)Google Scholar
  7. 7.
    Bostan, A., Raschel, K., Salvy, B.: Non-D-finite excursions in the quarter plane. J. Comb. Theory Ser. A 121, 45–63 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bousquet-Mélou, M.: Walks in the quarter plane: Kreweras’ algebraic model. Ann. Appl. Probab. 15(2), 1451–1491 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bousquet-Mélou, M.: An elementary solution of Gessel’s walks in the quadrant. (2015)
  10. 10.
    Bousquet-Mélou, M., Mishna, M.: Walks with small steps in the quarter plane. In: Algorithmic Probability and Combinatorics, vol. 520 of Contemporary Mathematics, pp. 1–40. American Mathematical Society, Providence, RI (2010)Google Scholar
  11. 11.
    Bousquet-Mélou, M., Petkovšek, M.: Walks confined in a quadrant are not always D-finite. Theor. Comput. Sci. 307(2):257–276. Random generation of combinatorial objects and bijective combinatorics (2003)Google Scholar
  12. 12.
    Christol, G.: Globally bounded solutions of differential equations. In: Analytic Number Theory (Tokyo, 1988), vol. 1434 of Lecture Notes in Mathematics, pp. 45–64. Springer, Berlin (1990)Google Scholar
  13. 13.
    Denisov, D., Wachtel, V.: Random walks in cones. Ann. Probab. 43(3), 992–1044 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fayolle, G., Iasnogorodski, R., Malyshev, V.: Random Walks in the Quarter-Plane. Algebraic Methods, Boundary Value Problems and Applications, vol. 40. Springer-Verlag, Berlin Heidelberg (1999)Google Scholar
  15. 15.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  16. 16.
    Garrabrant, S., Pak, I.: Counting with irrational tiles.
  17. 17.
    Gessel, I.M., Zeilberger, D.: Random walk in a Weyl chamber. Proc. Am. Math. Soc. 115(1), 27–31 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grabiner, D.J.: A combinatorial correspondence for walks in Weyl chambers. J. Comb. Theory Ser. A 71(2), 275–292 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grabiner, D.J., Magyar, P.: Random walks in Weyl chambers and the decomposition of tensor powers. J. Algebr. Comb. 2(3), 239–260 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Humphreys, J.E.: Introduction to Lie algebras and representation theory. In: Graduate Texts in Mathematics, vol. 9. Springer, New York (1972)Google Scholar
  21. 21.
    Janse van Rensburg, E.J., Prellberg, T., Rechnitzer, A.: Partially directed paths in a wedge. J. Comb. Theory Ser. A 115(4), 623–650 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kauers, M., Jaroschek, M., Johansson, F.: Computer Algebra and Polynomials. In: Gutierrez, J., Schicho, J., Weimann, M. (eds.) Ore Polynomials in Sage, vol. 8942, pp. 105–125. Springer International Publishing (2015)Google Scholar
  23. 23.
    Koutschan, C.: A fast approach to creative telescoping. Math. Comput. Sci. 4(2–3), 259–266 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lipshitz, L.: D-finite power series. J. Algebra 122(2), 353–373 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Melczer, S., Mishna, M.: Singularity analysis via the iterated kernel method. Comb. Probab. Comput. 23, 861–888 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pemantle, R., Wilson, M.C.: Asymptotics of multivariate sequences: I. Smooth points of the singular variety. J. Comb. Theory Ser. A 97(1), 129–161 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pemantle, R., Wilson, M.C.: Analytic Combinatorics in Several Variables. Cambridge University Press, Cambridge (2013)CrossRefzbMATHGoogle Scholar
  28. 28.
    Raichev, A.: Amgf documentation—release 0.8. (2012)
  29. 29.
    Raichev, A., Wilson, M.C.: Asymptotics of coefficients of multivariate generating functions: improvements for smooth points. Electr. J. Comb. 15(1) (2008)Google Scholar
  30. 30.
    Raichev, A., Wilson, M.C.: Asymptotics of coefficients of multivariate generating functions: improvements for multiple points. Online J. Anal. Comb. 6 (2011)Google Scholar
  31. 31.
    Wimp, J., Zeilberger, D.: Resurrecting the asymptotics of linear recurrences. J. Math. Anal. Appl. 111(1), 162–176 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Xin, G.: Determinant formulas relating to tableaux of bounded height. Adv. Appl. Math. 45(2), 197–211 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zeilberger, D.: A holonomic systems approach to special functions identities. J. Comput. Appl. Math. 32(3), 321–368 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.U. Lyon, CNRS, ENS de Lyon, INRIAUCBL, Laboratoire LIPLyonFrance
  3. 3.Department of MathematicsSimon Fraser UniversityBurnabyCanada

Personalised recommendations