Abstract
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.
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- 1.
By a Laurent polynomial in u we mean a polynomial in u and \(u^{-1}\).
- 2.
Here, by Laurent series we mean a series of the form \(H(z)=\sum _{n\ge a} H_n\,z^n\) for some (possibly negative) integer a.
- 3.
In this article, by convention \(0\in \mathbb {N}\).
- 4.
In this article, whenever we thought it could ease the reading, without harming the understanding, we write \(u_1\) for \(u_1(z)\), or F for F(z), etc.
- 5.
The quasi-inverse of a power series f(z) of positive valuation is \(1/(1-f(z))\).
- 6.
Unconstrained means that the walks are allowed to have both positive and negative altitudes.
- 7.
Axxxxxx refers to the corresponding sequence in the On-Line Encyclopedia of Integer Sequences, available electronically at https://oeis.org.
- 8.
Here, the \(^{*}\) is a mnemonic to remind us that we do not have the 0-step.
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Acknowledgements
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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Banderier, C. et al. (2019). Explicit Formulas for Enumeration of Lattice Paths: Basketball and the Kernel Method. In: Andrews, G., Krattenthaler, C., Krinik, A. (eds) Lattice Path Combinatorics and Applications. Developments in Mathematics, vol 58. Springer, Cham. https://doi.org/10.1007/978-3-030-11102-1_6
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