Counting Walks in the Quarter Plane

  • Mireille Bousquet-Mélou
Conference paper
Part of the Trends in Mathematics book series (TM)


We study planar walks that start from a given point (i0j0), take their steps in a finite set \(\mathfrak{S}\), and are confined in the first quadrant x ≥ 0, y ≥ 0. Their enumeration can be attacked in a systematic way: the generating function Q(x, y; t) that counts them by their length (variable t) and the coordinates of their endpoint (variables x, y) satisfies a linear functional equation encoding the step-by-step description of walks. For instance, for the square lattice walks starting from the origin this equation reads
$$\left( {xy - t(x + y + {x^2}y + x{y^2})} \right) {\text{Q(x,y;t)}} {\text{ = }} {\text{xy - xtQ(x,0;t) - ytQ(0,y;t)}}{\text{.}}$$

The central question addressed in this paper is the nature of the series Q(x, y ; t). When is it algebraic? When is it D-finite (or holonomic)? Can these properties be derived from the functional equation itself?

Our first result is a new proof of an old theorem due to Kreweras, according to which one of these walk models has, for mysterious reasons, an algebraic generating function. Then, we provide a new proof of a holonomy criterion recently proved by M. Petkovšek and the author. In both cases, we work directly from the functional equation.


Functional Equation Power Series Formal Power Series Positive Part Lattice Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    D. André, Solution directe du problème résolu par M. BertrandC.R. Acad. Sci. Paris105 (1887) 436–437.Google Scholar
  2. [2]
    C. Banderier, M. Bousquet-Mélou, A. Denise, D. Gardy, D. GouyouBeauchamps and P. Flajolet, Generating functions for generating trees Discrete Math. 246 No. 1–3 (2002) 29–55.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    C. Banderier and P. Flajolet, Basic analytic combinatorics of directed lattice paths, to appear in Theoret. Comput. Sci. Google Scholar
  4. [4]
    E. Barcucci, E. Pergola, R. Pinzani and S. Rinaldi, A bijection for some paths on the slit plane, Adv. in Appl. Math. 26 No. 2 (2001) 89–96.MathSciNetzbMATHGoogle Scholar
  5. [5]
    M. Bousquet-Mélou, Multi-statistic enumeration of two-stack sortable permutations Electron. J. Combin. 5 (1998) R21.Google Scholar
  6. [6]
    M. Bousquet-Mélou, Walks on the slit plane: other approaches, Adv. in Appl. Math. 27 No. 2–3 (2001) 243–288.zbMATHGoogle Scholar
  7. [7]
    M. Bousquet-Mélou, On (some) functional equations arising in enumerative combinatorics, in Proceedings of FPSAC’01 Arizona State University, pp. 83–89.Google Scholar
  8. [8]
    M. Bousquet-Mélou, Walks in the quarter plane: an algebraic example, in preparation.Google Scholar
  9. [9]
    M. Bousquet-Mélou, The number of vexillary involutions, in preparation.Google Scholar
  10. [10]
    M. Bousquet-Mélou and M. Petkovgek, Linear recurrences with constant coefficients: the multivariate case Discrete Math. 225 (2000) 51–75.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    M. Bousquet-Mélou and M. Petkovgek, Walks confined in a quadrant are not always D-finite, submitted.Google Scholar
  12. [12]
    M. Bousquet-Mélou and G. Schaeffer, Walks on the slit plane, preprint 2000, arXiv:math.CO/0012230. To appear in Proba. Theory Related Fields. Google Scholar
  13. [13]
    W. G. Brown, On the existence of square roots in certain rings of power series Math. Annalen 158 (1965) 82–89.zbMATHCrossRefGoogle Scholar
  14. [14]
    P. Duchon, On the enumeration and generation of generalized Dyck words Discrete Math. 225 (2000) 121–135.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    G. Fayolle, R. Iasnogorodski and V. Malyshev Random walks in the quarter plane Vol. 40 of Applications of Mathematics: Stochastic Modelling and Applied Probability Springer-Verlag, Berlin Heidelberg, 1999.Google Scholar
  16. [16]
    W. Feller An introduction to probability theory and its applications Vol.l, John Wiley and Sons, New York, 1950.Google Scholar
  17. [17]
    P. Flajolet, Analytic models and ambiguity of context-free languages Theoret. Comput. Sci. 49 (1987) 283–309.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    P. Flajolet and A. Odlyzko, Singularity analysis of generating functions SIAM J. Discrete Math. 3 No. 2 (1990) 216–240.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    L. Flatto and S. Hahn, Two parallel queues created by arrivals with two demands I. SIAM J. Appl. Math. 44 (1984) 1041–1053.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    I. M. Gessel, A probabilistic method for lattice path enumeration J. Stat. Planning Infer. 14 (1986) 49–58.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    I.M. Gessel, A factorization for formal Laurent series and lattice path enumeration J. Combin. Theory. Ser. A 28 (1980) 321–337.MathSciNetCrossRefGoogle Scholar
  22. [22]
    I. M. Gessel and D. Zeilberger, Random walk in a Weyl chamber Proc. Amer. Math. Soc. 115 (1992) 27–31.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    I. P. Goulden and D. M. Jackson Combinatorial enumeration John Wiley and Sons, 1983.Google Scholar
  24. [24]
    D. J. Grabiner, Random walk in an alcove of an affine Weyl group, and non-colliding random walks on an interval, in Proceedings of FPSAC 2001 Tempe, Arizona, pp. 217–225.Google Scholar
  25. [25]
    R. Guy, B. E. Sagan and C. Krattenthaler, Lattice paths, reflections, and dimension-changing bijections Ars Combin. 34 (1992) 3–15.MathSciNetzbMATHGoogle Scholar
  26. [26]
    O. Guibert, E. Pergola and R. Pinzani, Vexillary involutions are enumerated by Motzkin numbers Ann. Comb. 5 (2001) 1–22.MathSciNetCrossRefGoogle Scholar
  27. [27]
    G. Kreweras, Sur une classe de problèmes liés au treillis des partitions d’entiers Cahiers du B.U.R.O. 6 (1965) 5–105.Google Scholar
  28. [28]
    J. Labelle, Langages de Dyck généralisés, Ann. Sci. Math. Québec 17 (1993) 53–64.MathSciNetzbMATHGoogle Scholar
  29. [29]
    J. Labelle and Y.-N. Yeh, Generalized Dyck paths Discrete Math. 82 (1990) 1–6.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    G. F. Lawler Intersections of random walks Probabilities and its applications, Birkhäuser Boston, 1991.zbMATHCrossRefGoogle Scholar
  31. [31]
    L. Lipshitz, The diagonal of a D-finite power series is D-finite J. Algebra 113 (1988) 373–378.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    L. Lipshitz, D-finite power series J. Algebra 122 (1989) 353–373.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    S. G. Mohanty Lattice path counting and applications Academic Press, 1979.Google Scholar
  34. [34]
    T. V. Narayana, A partial order and its applications to probability theory Sankhya 21 (1959) 91–98.MathSciNetzbMATHGoogle Scholar
  35. [35]
    H. Niederhausen, The ballot problem with three candidates European J. Combin. 4 (1983) 161–167.MathSciNetzbMATHGoogle Scholar
  36. [36]
    H. Niederhausen, Lattice paths between diagonal boundaries Electron. J. Combin. 5 (1998) R30.MathSciNetGoogle Scholar
  37. [37]
    D. Poulalhon and G. Schaeffer, A bijection for loopless triangulations of a polygon with interior points, FPSAC 2002.Google Scholar
  38. [38]
    G. Schaeffer, Random sampling of large planar maps and convex polyhedra, in Proceedings of the 31th annual ACM Symposium on the Theory of Computing (STOC’99), Atlanta, ACM press, 1999.Google Scholar
  39. [39]
    F. Spitzer Principles of random walk The University Series in Higher Mathematics, Van Nostrand Company, Princeton, 1964.Google Scholar
  40. [40]
    R. P. Stanley, Differentiably finite power series European J. Combin. 1 (1980) 175–188.MathSciNetzbMATHGoogle Scholar
  41. [41]
    W. T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962) 21–38.MathSciNetzbMATHCrossRefGoogle Scholar
  42. [42]
    W. T. Tutte, A census of planar maps, Canad. J. Math.15 (1963) 249–271.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    J. West, Sorting twice through a stack, Theoret. Comput. Sci. 117 (1993) 303–313.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    D. Zeilberger, A proof of Julian West’s conjecture that the number of twostack-sortable permutations of length n is 2(3n)!/((n + 1)!(2n + 1)!), Discrete Math. 102 (1992) 85–93.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [45]
    D. Zeilberger, The umbral transfer-matrix method: I. Foundations, J. Combin. Theory Ser. A 91 (2000) 451–463.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Mireille Bousquet-Mélou
    • 1
  1. 1.CNRS, LaBRIUniversité BordeauxTalence CedexFrance

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