Unimodality, Log-concavity, Real-rootedness And Beyond

Krattenthaler, Christian

doi:10.1201/b18255-13

Cited by 26 publications

(12 citation statements)

References 107 publications

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“…Many efforts have been deployed in recent years for classifying them, see e.g. the surveys [27] and [30] and the references therein. The generating functions of lattice walks are not only intriguing for combinatorial reasons, but also from the perspective of computer algebra.…”

Section: Introductionmentioning

confidence: 99%

Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

Bostan

Chyzak

Hoeij

et al. 2017

European Journal of Combinatorics

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Abstract. We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on Z 2 defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or −1. We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane , pp. 201-215, 2009]. We give here the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions that are intimately related to elliptic integrals. As a second corollary, we show that all the 19 generating functions are transcendental, and that among their 19 × 4 combinatorially meaningful specializations only four are algebraic functions.

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Section: Introductionmentioning

confidence: 99%

Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

Bostan

Chyzak

Hoeij

et al. 2017

European Journal of Combinatorics

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“…3. A sequence of K + 1 points in Z M +1 is called lattice path of K steps [18]. Random walks over sites of Simp (N ) Z M +1 are defined by a set of admissible steps Ω M (step set Ω M ) so that at each step an i th coordinate m i increases by unity, while the nearest neighbouring one decreases by unity.…”

Section: It Satisfies the Equationsmentioning

confidence: 99%

“…Different types of random walks [14,18,31,32] are of considerable recent interest due to their role in quantum information processing [10,29]. The walks on multi-dimensional lattices were studied by many authors [4,23,24,28].…”

Section: Introductionmentioning

confidence: 99%

Zero Range Process and Multi-Dimensional Random Walks

Bogoliubov¹,

Malyshev²

2017

SIGMA

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Abstract. The special limit of the totally asymmetric zero range process of the low-dimensional non-equilibrium statistical mechanics described by the non-Hermitian Hamiltonian is considered. The calculation of the conditional probabilities of the model are based on the algebraic Bethe ansatz approach. We demonstrate that the conditional probabilities may be considered as the generating functions of the random multi-dimensional lattice walks bounded by a hyperplane. This type of walks we call the walks over the multi-dimensional simplicial lattices. The answers for the conditional probability and for the number of random walks in the multi-dimensional simplicial lattice are expressed through the symmetric functions.

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“…For the enumeration of simple lattice paths (allowing just the jumps −1, 0, and +1), many methods are often used, like e.g. the Lagrange inversion, determinant techniques, continued fractions, orthogonal polynomials, bijective proofs, and a lot is known in such cases [32,45,52,54]. These nice methods do not apply to more complex cases of more generic jumps (or, if one adds a spacial boundary, like a line of rational slope).…”

Section: Introductionmentioning

confidence: 99%

The Kernel Method for Lattice Paths Below a Line of Rational Slope

Banderier¹,

Wallner²

2019

Lattice Path Combinatorics and Applications

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We dedicate this article to the memory of Philippe Flajolet, who was and will remain a guide and a wonderful source of inspiration for so many of us. UUU AbstractWe analyse some enumerative and asymptotic properties of lattice paths below a line of rational slope. We illustrate our approach with Dyck paths under a line of slope 2/5. This answers Knuth's problem #4 from his "Flajolet lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in June 2014. Our approach extends the work of Banderier and Flajolet for asymptotics and enumeration of directed lattice paths to the case of generating functions involving several dominant singularities, and has applications to a full class of problems involving some "periodicities".A key ingredient in the proof is the generalization of an old trick by Knuth himself (for enumerating permutations sortable by a stack), promoted by Flajolet and others as the "kernel method". All the corresponding generating functions are algebraic, and they offer some new combinatorial identities, which can also be tackled in the A=B spirit of Wilf-Zeilberger-Petkovšek.We show how to obtain similar results for any rational slope. An interesting case is e.g. Dyck paths below the slope 2/3 (this corresponds to the so-called Duchon's club model), for which we solve a conjecture related to the asymptotics of the area below such lattice paths. Our work also gives access to lattice paths below an irrational slope (e.g. Dyck paths below y = x/ √ 2), a problem that we study in a companion article.Lattice paths below a line of rational slope 1 The "kernel method" that we mention here for functional equations in combinatorics has nothing to do with what is known as the "kernel method" or "kernel trick" in statistics or machine learning. Also, there is no integral directly related to our kernel. For sure, in our case the word kernel was chosen as its zeros will play a key role, and also, in one sense, as this kernel has in its core the full description of the problem, and its resolution.

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Unimodality, Log-concavity, Real-rootedness And Beyond

Cited by 26 publications

References 107 publications

Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

Zero Range Process and Multi-Dimensional Random Walks

The Kernel Method for Lattice Paths Below a Line of Rational Slope

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