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Asymptotic Lattice Path Enumeration Using Diagonals

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Abstract

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.

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Notes

  1. A function is D-finite if it satisfies a linear differential equation with polynomial coefficients.

  2. The walks themselves are not required to possess any particular kind of symmetry.

  3. The class of D-finite functions is closed under the diagonal operation  [24].

  4. In general, the kernel method associates to each set of steps \(\mathcal {S}\subset \{\pm 1,0\}^d\) some (possibly unique) group of rational maps (which groups arise, and what properties they possess, is still being investigated). When this group is finite an analysis similar to the one presented here often, but not always, yields an explicit expression for the corresponding generating function, and proves that this generating function is D-finite. See [10] for the two dimensional case and [3] for some results in three dimensions.

  5. Theorem 3.5 of [26] allows for asymptotic expansions of coefficient sequences more generally defined from multivariate functions than the diagonal sequence. The formula listed in Theorem 3.3 is stated only for diagonal sequences, as are our definitions of critical and minimal points.

  6. The input is formatted for Maple version 18.

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Acknowledgments

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.

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Authors and Affiliations

  1. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada

    Stephen Melczer

  2. U. Lyon, CNRS, ENS de Lyon, INRIA, UCBL, Laboratoire LIP, Lyon, France

    Stephen Melczer

  3. Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

    Marni Mishna

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  1. Stephen Melczer
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  2. Marni Mishna
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Correspondence to Marni Mishna.

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Melczer, S., Mishna, M. Asymptotic Lattice Path Enumeration Using Diagonals. Algorithmica 75, 782–811 (2016). https://doi.org/10.1007/s00453-015-0063-1

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