Counting Lattice Paths via a New Cycle Lemma

Nakamigawa, Tomoki; Tokushige, Norihide

doi:10.1137/100796431

Cited by 5 publications

(8 citation statements)

References 6 publications

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“…In this section, we show the connection between a result of Nakamigawa and Tokushige [53] and Knuth's statement. Furthermore, we derive extensions of this result.…”

Section: Links With the Work Of Nakamigawa And Tokushigementioning

confidence: 73%

“…The title of his lecture was "Problems that Philippe would have loved" and he was pinpointing/developing five nice open problems with a good flavor of "analytic combinatorics" (his slides are available online 2 ). The fourth problem was on "Lattice paths of slope 2/5", in which Knuth investigated Dyck paths under a line of slope 2/5, following the work of [53]. This is best summarized by the two following original slides of Knuth:…”

Section: Knuth's Aofa Problem #4mentioning

confidence: 99%

“…In the article[53], Q = (m, n); we changed these coordinates in order to avoid a conflict with our other notations.…”

mentioning

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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“…In this section, we show the connection between a result of Nakamigawa and Tokushige [53] and Knuth's statement. Furthermore, we derive extensions of this result.…”

Section: Links With the Work Of Nakamigawa And Tokushigementioning

confidence: 73%

Section: Knuth's Aofa Problem #4mentioning

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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“…The following result is well-known, see, e.g., Exercise 5.3.5 (b) in [5], and [8] for some extensions.…”

Section: Preliminariesmentioning

confidence: 96%

The maximum measure of 3-wise t-intersecting families

Tokushige¹

2021

Preprint

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Let G be a family of subsets of an n-element set. The family G is called 3-wise t-intersecting if the intersection of any three subsets in G is of size at least t. For a real number p ∈ (0, 1) we define the measure of the family by the sum of p |G| (1 − p) n−|G| over all G ∈ G. We prove that if p ≤ 2/( √ 4t + 9 − 1) then p t is the maximum measure of 3-wise t-intersecting families, and the bound for p is sharp. We also present the corresponding stability result.

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“…It is worth mentioning that Irving and Rattan [11,Theorem 1] gave a more general result, which was essentially equivalent to a conjecture by Tamm [22], and also generalized some recent work [2,8] on the enumeration of lattice paths. Moreover, Nakamigawa and Tokushige [19] gave a generalization of [11,Theorem 1] via a new cycle lemma.…”

Section: Introductionmentioning

confidence: 99%

A Chung–Feller theorem for lattice paths with respect to cyclically shifting boundaries

Guo

Wang

2018

J Algebr Comb

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Irving and Rattan gave a formula for counting lattice paths dominated by a cyclically shifting piecewise linear boundary of varying slope. Their main result may be considered as a deep extension of well-known enumerative formulas concerning lattice paths from (0, 0) to (kn, n) lying under the line x = ky (e.g., the Dyck paths when k = 1). On the other hand, the classical Chung-Feller theorem tells us that the number of lattice paths from (0, 0) to (n, n) with exactly 2k steps above the line x = y is independent of k and is therefore the Catalan number 1 n+1 2n n . In this paper, we study the number of lattice path boundary pairs (P, a) with k flaws, where P is a lattice path from (0, 0) to (n, m), a is a weak m-part composition of n, and a flaw is a horizontal step of P above the boundary ∂a. We prove bijectively, for a given a, that summing these numbers over all cyclic shifts of the boundary ∂a is equal to n+m m−1 . That is, we generalize the Irving-Rattan formula to a Chung-Feller-type theorem. We also give a refinement of this result by taking the number of double ascents of lattice paths into account.

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Counting Lattice Paths via a New Cycle Lemma

Cited by 5 publications

References 6 publications

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

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

The maximum measure of 3-wise t-intersecting families

A Chung–Feller theorem for lattice paths with respect to cyclically shifting boundaries

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