Explicit Formulas for Enumeration of Lattice Paths: Basketball and the Kernel Method

  • Cyril Banderier
  • Christian Krattenthaler
  • Alan Krinik
  • Dmitry Kruchinin
  • Vladimir Kruchinin
  • David Nguyen
  • Michael WallnerEmail author
Part of the Developments in Mathematics book series (DEVM, volume 58)


This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude j and ending at a given altitude k, with additional constraints, for example, to never attain altitude 0 in-between. We first discuss the case of walks on the integers with steps \(-h, \dots , -1, +1, \dots , +h\). The case \(h=1\) is equivalent to the classical Dyck paths, for which many ways of getting explicit formulas involving Catalan-like numbers are known. The case \(h=2\) corresponds to “basketball” walks, which we treat in full detail. Then, we move on to the more general case of walks with any finite set of steps, also allowing some weights/probabilities associated with each step. We show how a method of wide applicability, the so-called kernel method, leads to explicit formulas for the number of walks of length n, for any h, in terms of nested sums of binomials. We finally relate some special cases to other combinatorial problems, or to problems arising in queuing theory.


Lattice paths Dyck paths Motzkin paths Kernel method Analytic combinatorics Computer algebra Generating function Singularity analysis Lagrange inversion Context-free grammars \(\mathbb {N}\)-algebraic function 

2010 Mathematics Subject Classification

Primary: 05A15 Secondary: 05A10 05A16 05A19 



We thank the organizers of the 8th International Conference on Lattice Path Combinatorics & Applications, which provided the opportunity for this collaboration. Sri Gopal Mohanty played an important role in the birth of this sequence of conferences, and his book [42] was the first one (together with the book of his Ph.D. advisor Tadepalli Venkata Narayana [44]) to spur strong interest in lattice path enumeration. We are therefore pleased to dedicate our article to him.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Cyril Banderier
    • 1
  • Christian Krattenthaler
    • 2
  • Alan Krinik
    • 3
  • Dmitry Kruchinin
    • 4
  • Vladimir Kruchinin
    • 4
  • David Nguyen
    • 5
  • Michael Wallner
    • 6
    • 7
    Email author
  1. 1.Laboratoire d’Informatique de Paris-NordCNRS/UniversitéVilletaneuseFrance
  2. 2.Fakultät für MathematikUniversität WienViennaAustria
  3. 3.Department of Mathematics and StatisticsCalifornia State Polytechnic University, PomonaPomonaUSA
  4. 4.Tomsk State University of Control Systems and Radio ElectronicsTomskRussia
  5. 5.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA
  6. 6.Institute of Discrete Mathematics and GeometryTU WienAustria
  7. 7.Laboratoire Bordelais de Recherche en InformatiqueUniversité de BordeauxBordeauxFrance

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