Combinatorics of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml371" display="inline" overflow="scroll" altimg="si7.gif"><mml:mi>n</mml:mi></mml:math>-color cyclic compositions

Gibson, Meghann Moriah; Gray, Daniel; Wang, Hua

doi:10.1016/j.disc.2018.08.001

Cited by 4 publications

(6 citation statements)

References 7 publications

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“…Finally, we consider the class of n-colour compositions, introduced by Agarwal [1] and studied in Gibson et al [27], where each integer i is coloured by one of i available colours (each summand must be coloured and different colourings give distinct n-colour compositions).…”

Section: N-colour Compositions With Tracked Summandsmentioning

confidence: 99%

“…The authors of this paper were introduced to Conjecture 21 by an author of Chung et al[10], who lamented on the opacity of some past references using ACSV to establish limit theorems.the electronic journal of combinatorics 31(2) (2024), #P2 27. …”

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confidence: 99%

“…For instance, Pemantle et al[45] includes roughly 100 pages of appendices with (highly abridged) background material, beyond the additional background material discussed in the main text.the electronic journal of combinatorics 31(2) (2024), #P2 27. …”

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confidence: 99%

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Central Limit Theorems via Analytic Combinatorics in Several Variables

Melczer,

Ruza

2024

Electron. J. Combin.

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The field of analytic combinatorics is dedicated to the creation of effective techniques to study the large-scale behaviour of combinatorial objects. Although classical results in analytic combinatorics are mainly concerned with univariate generating functions, over the last two decades a theory of analytic combinatorics in several variables (ACSV) has been developed to study the asymptotic behaviour of multivariate sequences. In this work we survey ACSV from a probabilistic perspective, illustrating how its most advanced methods provide efficient algorithms to derive limit theorems, and comparing the results to past work deriving limit theorems. Using the results of ACSV, we provide a SageMath package that can automatically compute (and rigorously verify) limit theorems for a large class of combinatorial generating functions. To illustrate the techniques involved, we also establish an explicit local central limit theorems for a family of combinatorial classes whose generating functions are linear in the variables tracking each parameter. Applications covered by this result include the distribution of cycles in certain restricted permutations (proving a limit theorem stated as a conjecture in recent work of Chung et al.), integer compositions, and n-colour compositions with varying restrictions and values tracked. Key to establishing these explicit results in an arbitrary dimension is an interesting symbolic determinant, which we compute by conjecturing and then proving an appropriate LU-factorization. It is our hope that this work provides readers a blueprint to apply the powerful tools of ACSV in their own work, making them more accessible to combinatorialists, probabilists, and those in adjacent fields.

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Section: N-colour Compositions With Tracked Summandsmentioning

confidence: 99%

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confidence: 99%

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Central Limit Theorems via Analytic Combinatorics in Several Variables

Melczer,

Ruza

2024

Electron. J. Combin.

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“…Next, we prove a connection between n-color compositions prohibiting the color 2 and certain regular compositions. To present this bijection, we introduce the (5,7,6) where 5 is a result of 3· and ·2·, 7 is a result of 4· and ·3, and 6 is a result of ·1· and ·5; see Figure 8. Figure 8.…”

Section: 22mentioning

confidence: 99%

“…Spotted tilings for the n-color compositions of 3 are shown in Figure 1. Spotted tiling were subsequently used in [6,7]. Other combinatorial interpretations of n-colored compositions are based on binary sequences [15], lattice paths [3], and rooted trees [14].…”

Section: Introductionmentioning

confidence: 99%

Chapter 6: Spotted Tilings and n-Color Compositions

Hopkins¹

2013

Combinatorial Number Theory

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Agarwal introduced n-color compositions in 2000 and most subsequent research has focused on restricting which parts are allowed. Here we focus instead on restricting allowed colors. After three general results, giving recurrence formulas for the cases of given allowed colors, given prohibited colors, and colors satisfying modular conditions, we consider several more specific conditions, establishing direct formulas and connections to other combinatorial objects. Proofs are combinatorial, mostly using the notion of spotted tilings introduced by the first named author in 2012.

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Generalized colored circular palindromic compositions

Hadjicostas

2020

Moscow J. Comb. Number Th.

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Combinatorics of n-color cyclic compositions

Cited by 4 publications

References 7 publications

Central Limit Theorems via Analytic Combinatorics in Several Variables

Central Limit Theorems via Analytic Combinatorics in Several Variables

Chapter 6: Spotted Tilings and n-Color Compositions

Generalized colored circular palindromic compositions

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