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

Banderier, Cyril; Krattenthaler, Christian; Krinik, Alan C.; Kruchinin, Dmitry; Kruchinin, Vladimir; Nguyen, David Tuan; Wallner, Michael

doi:10.1007/978-3-030-11102-1_6

Cited by 17 publications

(21 citation statements)

References 32 publications

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“…Also it should be noted that by using the proposed approach P. Barry [11] found formulas for the central coefficients of Riordan matrices and Cyril Banderier and other [12] found explicit formulas for enumeration of lattice paths.…”

Section: Lemma 2 Supposementioning

confidence: 94%

New Approach to Study Numeric Triangles

Kruchinin¹

2018

KEG

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In this paper we deal with numerical triangles defined by generating functions in the power of k. We present new approach to study such triangles that allow us to get methods for obtaining generating functions of the diagonals of the triangles. Methods for obtaining generating functions of the central coefficients and the diagonal T 2 , of the triangle T , are discussed. Some further ideas for application are given. Publishing services provided by Knowledge E Dmitry V Kruchinin. This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited. Selection and Peer-review under the responsibility of the RFYS Conference Committee.

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Section: Lemma 2 Supposementioning

confidence: 94%

New Approach to Study Numeric Triangles

Kruchinin¹

2018

KEG

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“…We want to end this chapter with exemplifying the miracles involved in the simplifications of (11). Using the Flajolet-Soria formula [7] for the coefficients of an algebraic function, we can extract the coefficient of z 7n−2 of G 1 (z) and F 0 (z) in terms of nested sums.…”

Section: Some Additional Investigations Conducted By Manuel Kauers (Pmentioning

confidence: 99%

“…Lattice paths below a line of rational slope (2,5,4,7,6,9,12,8,11,14,10,13,16,19,12,15,18,14,17,20,16,19,22,18,21,24,20,23,26,22,25,24,27,26,29,28,31,30,33,32,34,36,38,40) and (c n ) 44 n=1 = (2,0,3,1,4,2,0,5,3,1,6,4,2,0,7,5, 3,8,6,4,9,7,5,10,…”

Section: Cyril Banderier and Michael Wallnermentioning

confidence: 99%

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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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“…where D(t) := [u 0 ]u e K (t, u) is either some power of t, or a difference of two powers of t (similarly to (11), but with more cases that will be specified in the proof in Sect. 5).…”

Section: The Bivariate Generating Function For Meanders Avoiding the mentioning

confidence: 99%

Analytic Combinatorics of Lattice Paths with Forbidden Patterns, the Vectorial Kernel Method, and Generating Functions for Pushdown Automata

et al. 2019

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In this article we develop a vectorial kernel method-a powerful method which solves in a unified framework all the problems related to the enumeration of words generated by a pushdown automaton. We apply it for the enumeration of lattice paths that avoid a fixed word (a pattern), or for counting the occurrences of a given pattern. We unify results from numerous articles concerning patterns like peaks, valleys, humps, etc., in Dyck and Motzkin paths. This refines the study by Banderier and Flajolet from 2002 on enumeration and asymptotics of lattice paths: we extend here their results to pattern-avoiding walks/bridges/meanders/excursions. We show that the autocorrelation polynomial of this forbidden pattern, as introduced by Guibas and Odlyzko in 1981 in the context of rational languages, still plays a crucial role for our algebraic languages. En passant, our results give the enumeration of some classes of self-avoiding walks, and prove several conjectures from the On-Line Encyclopedia of Integer Sequences. Finally, we also give the trivariate generating function (length, final altitude, number of occurrences of the pattern p), and we prove that the number of occurrences is normally distributed and linear with respect to the length of the walk: this is what Flajolet and Sedgewick call an instance of Borges's theorem. Keywords Lattice paths • Dyck paths • Motzkin paths • Łukasiewicz paths • Pattern avoidance • Autocorrelation • Finite automata • Markov chains • Pushdown automata • Generating functions • Wiener-Hopf factorization • Kernel method • Asymptotic analysis • Gaussian limit law • Borges' theorem We dedicate this article to the memory of Philippe Flajolet, our cheerful and inspiring mentor, founder of analytic combinatorics.

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

Cited by 17 publications

References 32 publications

New Approach to Study Numeric Triangles

New Approach to Study Numeric Triangles

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

Analytic Combinatorics of Lattice Paths with Forbidden Patterns, the Vectorial Kernel Method, and Generating Functions for Pushdown Automata

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