Science topics: MathematicsCombinatorics
Science topic
Combinatorics - Science topic
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics).
Publications related to Combinatorics (10,000)
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This book is the fourth volume in the series Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond.
This volume specifically delves into the concept of the HyperUncertain Set, building on the foundational advancements introduced in previous volume...
This book is the fourth volume in the series Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. This volume specifically delves into the concept of the HyperUncertain Set, building on the foundational advancements introduced in previous volume...
A brief solution to the binomial option pricing models using combinatorics.
We discuss some combinatorics associated with 1-away permutations, where an element can be displaced from its correct position by at most one location. Specifically, we look at a sorting algorithm for such permutations and analyze its number of comparisons, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts...
Background
Comprehensive environmental risk characterization, encompassing physical, chemical, social, ecological, and lifestyle stressors, necessitates innovative approaches to handle the escalating complexity. This is especially true when considering individual and population-level diversity, where the myriad combinations of real-world exposures...
The primary objective of this study is the computation of the matching polynomials of a number of symmetric, semisymmetric, double group graphs, and solids in third and higher dimensions. Such computations of matching polynomials are extremely challenging problems due to the computational and combinatorial complexity of the problem. We also conside...
We introduce a new topological invariant of complex line arrangements in $\mathbb{CP}^2$, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski pairs which have the same combinatorics but different embeddings. Building on ideas developed by B. Guerville-Ball\'e and W...
We prove new quantitative bounds on the additive structure of sets obeying an $L^3$ 'control' assumption, which arises naturally in several questions within additive combinatorics. This has a number of applications - in particular we improve the known bounds for the sum-product problem, the Balog-Szemer\'{e}di-Gowers theorem, and the additive growt...
We investigate the combinatorics of permutations underlying the the daily word game Waffle, and learn why some games are easy to solve while extreme games are very hard. A perfect unscrambling must have precisely 11 orbits, with at least one of length 1, on the 21 squares.
Texas Hold'em is a skill and strategy game that combines elements of probability, combinatorics, and game theory. This paper explores the mathematical principles behind the game, with a focus on applying probability theory to decision-making and strategy development. Analyzing the win rates of various hand rankings, the expected value of different...
Una de las grandes tendencias de los dos últimos años, con merecida razón, ha sido la inteligencia artificial. Dos de los conceptos fundamentales sobre los cuales están creados casi todos los sistemas a los que hoy damos este nombre son las neuronas artificiales (más específicamente, los denominados perceptrón) y las redes neuronales (los perceptro...
The interval poset of a permutation is the set of intervals of a permutation, ordered with respect to inclusion. It has been introduced and studied recently in Tenner (Order 39(3), 523–536 2022). We study this poset from the perspective of the decomposition trees of permutations, describing a procedure to obtain the former from the latter. We then...
We present an algorithm for sampling tightly confined random equilateral closed polygons in three-space which has runtime linear in the number of edges. Using symplectic geometry, sampling such polygons reduces to sampling a moment polytope, and in our confinement model this polytope turns out to be very natural from a combinatorial point of view....
We study subsets of \(\mathbb {F}_p^n\) that do not contain progressions of length \(k\). We denote by \(r_k(\mathbb {F}_p^n)\) the cardinality of such subsets containing a maximal number of elements. In this paper we focus on the case \(k=p\) and therefore sets containing no full line. A trivial lower bound \(r_p(\mathbb {F}_p^n)\ge (p-1)^n\) is a...
Motivated by the definition of Freiman homomorphism we explore the possibilities of formulating some basic notions and techniques of additive combinatorics in a categorical language. We show that additive sets and Freiman homomorphisms form a category and we study several limit and colimit constructions in this, and in an interesting subcategory of...
Il contributo si concentra su un aspetto lessicale della lingua accademica italiana parlata di particolare rilievo per gli apprendenti stranieri, ovvero quello delle collocazioni. Difatti, la loro conoscenza e il corretto impiego di tali unità fraseologiche sono indice di un livello avanzato di competenza linguistico-comunicativa del parlante e del...
Este trabalho trata de um relato de experiência, ao qual visa trazer informações sobre uma
atividade realizada no Ensino Médio na aula de Matemática sobre polinômios geradores. Assim,
buscou-se reunir informações com o propósito de responder à seguinte questão: De que forma o
estudo sobre Funções Geradoras no Ensino Médio pode proporcionar difer...
bstract. Reresentations are prime in physics. Gallileo gets the biggest credit because he created the paradigm, using geometrical representations for motion and acceleration. Newton used the vector and point mass representation. Once thought as the ingenious attempt to copy Einstein's use of novel mathematical representations for gravity, i.e. non-...
Combinatorics is a branch of mathematics focused on counting, arranging, and combining elements within a set under specific rules and constraints. This field is particularly fascinating due to its ability to yield novel results through the integration of concepts from various mathematical domains. Its significance remains unchanged in areas that ad...
This paper introduces a novel representation of factorials as a product of sums of consecutive integers. A formal theorem is presented to establish this representation, along with detailed proofs and examples. The approach provides new insights into factorials, uncovering their structural patterns and offering potential applications in combinatoric...
Combinatorics is a branch of mathematics focused on counting, arranging, and combining elements within a set under specific rules and constraints. This field is particularly fascinating due to its ability to yield novel results through the integration of concepts from various mathematical domains. Its significance remains unchanged in areas that ad...
The summation of powers is a classical problem in mathematics, with particular interest in the recursive structures that govern the cumulative and decumulative frequencies of power series. In this paper, we introduce a novel framework where x represents the cumulative frequency degree, which behaves differently depending on the sign of x. Specifica...
We analyze the numbers of closed paths of length k∈N on two important regular lattices: the hexagonal lattice (also called graphene in chemistry) and its dual triangular lattice. These numbers form a moment sequence of specific random variables connected to the distance of a position of a planar random flight (in three steps) from the origin. Here,...
Vertically parametrised polynomial systems are a particular nice class of parametrised polynomial systems for which a lot of interesting algebraic information is encoded in its combinatorics. Given a fixed polynomial system, we empirically study what constitutes a good vertically parametrised polynomial system that gives rise to it and how to const...
Let G be a finite simple graph and let A(G) be its adjacency matrix. Then, G is singular if A(G) is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. For positive integers ai≥2, i=1,2,…,6. Insert a1−2, a2−2, a3−2, a4−2, a5−2 and a6−2 vertices in the six edges of the complete graph K4, respective...
Let $ (\rho, V) $ be an irreducible representation of the symmetric group $ S_n$ (or the alternating group $ A_n$), and let $ g $ be a permutation on $n$ letters with each of its cycle lengths divides the length of its largest cycle. We describe completely the minimal polynomial of $\rho(g)$, showing that, in most cases, it equals $x^{o(g)} - 1 $,...
I will show that there exist two binary words (one of length 4 and one of length 6) that play a special role in many different problems in combinatorics on words. They can therefore be considered \textit{the shortest interesting binary words}. My claim is supported by the fact that these two words appear in dozens of papers in combinatorics on word...
The objective of the max-cut problem is to cut any graph in such a way that the total weight of the edges that are cut off is maximum in both subsets of vertices that are divided due to the cut of the edges. Although it is an elementary graph partitioning problem, it is one of the most challenging combinatorial optimization-based problems, and tons...
Circuit algebras, used in the study of finite-type knot invariants, are a symmetric analogue of Jones's planar algebras. This work, the first of a pair of papers comprising a detailed study of circuit algebra combinatorics, provides three equivalent descriptions of circuit algebras: in terms of operads of wiring diagrams, modular operads and catego...
Circuit algebras originated in quantum topology and are a symmetric analogue of Jones's planar algebras. This paper is the second of a pair that together provide detailed conceptual and technical study of circuit algebra combinatorics. Extending existing results for modular operads, a graphical calculus and monad for circuit algebras is established...
Given that cortical manifolds in neural combing is is irregular in the mysthenia graves brain, we yield invariant signatures with continuable fractions for pharmocokinetase for regularizing gravis and obviating facial palsy w/wo neural circuitry. This is one of the unique techniques with bearings on statistical distributions in drug trials in a dua...
Combinatorial signatures of manifolds in relaxation states, a definitive cure for Mysthenia Gravis, Cortical simulation in combinatorics. Given that cortical manifolds in neural combing is is irregular in the mysthenia graves brain, we yield invariant signatures with continuable fractions for pharmocokinetase for regularizing gravis and obviating f...
We investigate the mutual relations between the centers of different elements in the deconstruction lattice of a 2D conformal model, and show how these can be described using exact sequences of abelian groups. In particular, we exhibit a long exact sequence connecting the centers of higher central quotients.
The combinatorics of dimer models on brane tilings describe a large class of four-dimensional $\mathcal{N}=1$ gauge theories that afford quiver descriptions and have toric moduli spaces. We introduce a combinatorial optimization method leveraging simulated annealing to explicitly construct geometrically consistent brane tilings, providing a proof o...
Cet ouvrage pédagogique est spécifiquement conçu pour les étudiants de deuxième année du parcours Ingénieur en Génie des Procédés. Conforme au canevas du parcours Ingénieur d’État, particulièrement destiné aux bacheliers TM, il propose une approche exhaustive et structurée des concepts essentiels de la statistique et des probabilités, indispensable...
In this article we compute and analyze the spectrum of operators defined by the metaplectic representation $\mu$ on the unitary group $\mathbb{U}(d)$ or operators defined by the corresponding induced representation $d\mu$ of the Lie algebra $\mathfrak{u}(d)$. It turns out that the point spectrum of both types of operators can be expressed in terms...
The graded M\"{o}bius algebra of a matroid is a commutative graded algebra which encodes the combinatorics of the lattice of flats of the matroid. As a special subalgebra of the augmented Chow ring of the matroid, it plays an important role in the recent proof of the Dowling-Wilson Top Heavy Conjecture. Recently, Mastroeni and McCullough proved tha...
If $\Lambda \subseteq \mathbb{Z}^n$ is a sublattice of index $m$, then $\mathbb{Z}^n/\Lambda$ is a finite abelian group of order $m$ and rank at most $n$. Several authors have studied statistical properties of these groups as we range over all sublattices of index at most $X$. In this paper we investigate quotients by sublattices that have addition...
Combinatorial properties of karyon tilings \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{T}$$\end{document} of torus \documentclass[12pt]{minimal} \usepackag...
Combinatorial optimization problems are considered to be an application, where quantum computing can have transformative impact. In the industrial context, job shop scheduling problems that aim at finding the optimal schedule for a set of jobs to be run on a set of machines are of immense interest. Here we introduce an efficient encoding of job sho...
En español y alemán, para expresar hiperbólicamente el hecho de experimentar una sensación física o psíquica se emplean las construcciones fraseológicas [V de (ART) Ssing{sensación}] / [vor NSg{Gefühlsempfindung} V], en las que destacan los verbos morir(se) y sterben por su alta frecuencia de uso y por su capacidad combinatoria, mucho mayor que la...
El artículo presenta una aproximación a la importancia de un enfoque integral para acompañar la configuración de proyectos de vida desarrolladores en los estudiantes, desde la perspectiva de la transformación organizacional y cultural en la Educación Superior. A través de una metodología combinatoria que integró la etnografía y la teoría fundamenta...
Perhaps the most familiar way of representing hierarchical structure in theories of natural language is by means of tree diagrams. These trees illustrate the derivations of sentences obtained by either top-down rewriting rules (as in Phrase Structure Grammars, expanding a sentential symbol) or bottom-up recursive discrete combinatorics (as in Minim...
This study presents the experience of calculating the differential features of substantive word forms that explicate in speech different degrees of functional and functional-semantic convergence with the class of adverbs of interval. Using the example of the instrumental case form poroj (sometimes), the combinatorics and proportion of features of n...
A$-type Little String Theories (LSTs) are engineered from parallel M5-branes on a circle $\mathbb{S}_\perp^1$, probing a transverse $\mathbb{R}^4/\mathbb{Z}_M$ background. Below the scale of the radius of $\mathbb{S}_\perp^1$, these theories resemble a circular quiver gauge theory with $M$ nodes of gauge group $U(N)$ and matter in the bifundamental...
In this work we fully characterize the classes of matrix weights for which multilinear Calder\'on-Zygmund operators extend to bounded operators on matrix weighted Lebesgue spaces. To this end, we develop the theory of multilinear singular integrals taking values in tensor products of finite dimensional Hilbert spaces. On the one hand, we establish...
A properly edge-colored graph is a graph with a coloring of its edges such that no vertex is incident to two or more edges of the same color. A subgraph is called rainbow if all its edges have different colors. The problem of finding rainbow subgraphs or other restricted structures in edge-colored graphs has a long history, dating back to Euler's w...
Neste trabalho estudamos o sistema fundamental generalizado de Jacobsthal, destacando suas principais propriedades e algumas identidades. São usadas as propriedades matriciais da bem conhecida matriz casoratiana para obtenção das identidades deste sistema. Também apresentamos expressões combinatórias para as propriedades e para as identidades. A me...
Reddit.com provides an excellent example of interactive digital communication. Reddit comments and posts organize information in line with certain rules of coding and decoding. They convey meanings by semiotically heterogeneous verbal and non-verbal means. The article describes various combinations of communication codes that act as a communication...
We introduce a super version of the Littlewood--Richardson rule for super Schur functions over signed alphabets.
We give in particular combinatorial interpretations of the super Littlewood--Richardson coefficients using the properties of super Young tableaux, which have found rich applications in representation theory, algebraic combinatorics, and...
These are notes of my lecture courses given in the summer of 2024 in the School on Number Theory and Physics at ICTP in Trieste and in the 27th Brazilian Algebra Meeting at IME-USP in S\~ao Paulo. We give an elementary account of $p$-adic methods in de Rham cohomology of algebraic hypersurfaces with explicit examples and applications in number theo...
Let $H_n^{(3)}$ be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, {\em wicket}, is formed by three rows and two columns of a $3 \times 3$ point matrix. In this note, we give a new lower bound on the Tur\'an number of wickets using estimates on cap sets. We also show that this problem is close...
The family of finite subsets $s$ of the natural numbers such that $|s|=1+\min s$ is known as the Schreier barrier in combinatorics and Banach Space theory, and as the family of exactly $\omega$-large sets in Logic. We formulate and prove the generalizations of Friedman's Free Set and Thin Set theorems and of Rainbow Ramsey's theorem to colorings of...
This study explored the hypothesis that modular and fractal patterns in Pascal’s Triangle correspond to human age-related developmental milestones. Pascal’s Triangle, known for its applications in combinatorics, reveals self-similar and fractal patterns, especially under modular transformations such as modulus 2 (forming the Sierpiński triangle). P...
In this paper we look at the ability of recent large language models (LLMs) at solving mathematical problems in combinatorics. We compare models LLaMA-2, LLaMA-3.1, GPT-4, and Mixtral against each other and against human pupils and undergraduates with prior experience in mathematical olympiads. To facilitate these comparisons we introduce the Combi...
This paper presents a recursive framework for analyzing cumulative power series using choice combinations and power coefficients. By focusing on the first (λ 1 = 1) and last (λ t) coefficients, and deriving intermediate coefficients (λ 2 ,. .. , λ t−1) algebraically , this approach simplifies the analysis of power series. For even powers, the choic...
We give a new construction of Lascoux-Sch\"uetzenberger's charge statistic in type A which is motivated by the geometric Satake equivalence. We obtain a new formula for the charge statistic in terms of modified crystal operators and an independent proof of this formula which does not rely on tableaux combinatorics.
Cet article étudie la combinatoire lexicale du mot embryon et la manière dont le sens de ce mot est profilé d’un syntagme à l’autre. L’étude a été menée à partir d’un corpus de bioéthique composé de discours institutionnels et de commentaires d’internautes. A partir de ce corpus, nous avons extrait à l’aide d’un concordancier tous les syntagmes dan...
![FIG. 2. The discrete-time signal x[n] is composed of sinusoidal...](publication/387078574/figure/fig1/AS:11431281298128630@1734318332405/The-discrete-time-signal-xn-is-composed-of-sinusoidal-components-with-frequencies-50_Q320.jpg)
This paper introduces a robust and scalable framework for implementing nested affine transformations in quantum circuits. Utilizing Hadamard-supported conditional initialization and block encoding, the proposed method systematically applies sequential affine transformations while preserving state normalization. This approach provides an effective m...
Background/objective
Preventive medications are crucial in migraine prevention. In cases of refractory migraine headaches, multiple medications may be required. We seek to identify a comprehensive list of preventive migraine headache medications that can be used as two, three, and four drug combinations without drug–drug interactions.
Methods
We c...
We study the interplay between Sabbah’s mixed Hodge structure for tame regular functions and Ehrhart theory for polytopes. We first analyze the Poincaré polynomial of the Hodge filtration of this mixed Hodge structure (we call this Poincaré polynomial the \(\theta \)-vector). Using the symmetry of the Hodge numbers involved, we show that it shares...
The Pixelverse represents a concept at the intersection of combinatorics, technology, and philosophy—a digital universe where every conceivable image can be generated by exhausting all possible pixel combinations on a screen. Rooted in the combinatorial theory similar to the Infinite Monkey Theorem, the Pixelverse theoretically contains every visua...
![Hook lengths for λ=(6,4,3,1)\documentclass[12pt]{minimal}...](publication/386875526/figure/fig1/AS:11431281296645818@1733971765705/Hook-lengths-for-l6-4-3-1documentclass12ptminimal-usepackageamsmath_Q320.jpg)
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Partition hook lengths have wide-ranging applications in combinatorics, number theory, physics, and representation theory. We study two infinite families of random variables associated with t-hooks. For fixed \(t\ge 1,\) if \(Y_{t;\,n}\) counts the number of hooks of length t in a random integer partition of n, we prove a uniform local limit theore...
Each hyperplane arrangement A in C^d gives rise to a Milnor fibration of its complement, F → M → C^*. Although the eigenvalues of the monodromy h: F → F acting on the homology groups H_i(F;C) can be expressed in terms of the jump loci for rank 1 local systems on M, explicit formulas are still lacking in full generality, even in degree i=1. In this...
A number of Erdélyi-type integrals for certain hypergeometric functions with their multidimensional extensions have been studied in the literature. The basic (or q-) analogs of such integrals have also been analyzed but the multidimensional extensions of basic (or q-) analogs of these integrals have not appeared so far. This work aims to develop mu...
Combinatorics is a branch of mathematics focused on counting, arranging, and combining elements within a set under specific rules and constraints. This field is particularly fascinating due to its ability to yield novel results through the integration of concepts from various mathematical domains. Its significance remains unchanged in areas that ad...
We introduce admissible Minkowski decomposition data (amd) for a 3-dimensional reflexive polytope P. This notion is defined purely in terms of the combinatorics of P. Denoting by X the Gorenstein toric Fano 3-fold whose fan is the spanning fan (a.k.a. face fan) of P, our first result states that amd for P determine a smoothing of X. Our second resu...
The second volume of "Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization" presents significant advancements in uncertain combinatorics through methodologies such as graphization and hyperization. It seamlessly integrates fuzzy, neutrosophic, soft, and rough set theories with combinatorics and graph theory to...
The third volume of “Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization” delves into groundbreaking developments in uncertain combinatorics and set theory. It highlights methodologies such as graphization, hyperization, and uncertainization, seamlessly integrating fuzzy, neutrosophic, soft, and rough set theo...
We discuss a link between graph theory and geometry that arises when considering graph dynamical systems with odd interactions. The equilibrium set in such systems is not a collection of isolated points, but rather a union of manifolds, which may intersect creating singularities and may vary in dimension. We prove that geometry and stability of suc...
This article introduces the 2-variable q-truncated exponential–Appell (q-trunc. exp. Appell) polynomials and investigates their fundamental properties. Specific results are derived for the q-trunc. exp. Appell family along with their graphical representations which contribute to advancing the understanding of q-series and q-special functions. Poten...
The second volume of "Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization" presents significant advancements in uncertain combinatorics through methodologies such as graphization and hyperization. It seamlessly integrates fuzzy, neutrosophic, soft, and rough set theories with combinatorics and graph theory to...
The third volume of “Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization” delves into groundbreaking developments in uncertain combinatorics and set theory. It highlights methodologies such as graphization, hyperization, and uncertainization, seamlessly integrating fuzzy, neutrosophic, soft, and rough set theo...
O presente artigo apresenta uma proposta de ensino decolonial que articula conceitos no campo da Matemática e História, lançando o jogo de búzios e as tradições culturais do candomblé como referências. O objetivo é promover uma reflexão sobre um modelo de ensino que valorize saberes marginalizados e reconheça a diversidade cultural afro-brasileira....
Several articles deal with tilings with various shapes, and also a very frequent type of combinatorics is to examine the walks on graphs or on grids. We combine these two things and give the numbers of the shortest walks crossing the tiled (1 × n ) and (2 × n ) square grids by covering them with squares and dominoes. We describe these numbers not o...
We present a library of formalized results around symmetric functions and the character theory of symmetric groups. Written in Coq/Rocq and based on the Mathematical Components library, it covers a large part of the contents of a graduate level textbook in the field. The flagship result is a proof of the Littlewood-Richardson rule, which computes t...
We examine students’ challenges in determining the number of distinct many-particle stationary states for a system of noninteracting identical particles, focusing on how these insights guided the design, validation, and evaluation of a quantum interactive learning tutorial (QuILT) to aid students’ understanding. Specifically, we focus on systems wi...
This research aims to analyze students' combinatorial thinking skills after using learning worksheets on enumeration rules. This research aims to provide solutions by applying learning materials to improve students' combinatorial thinking skills in solving math problems. The research method used is qualitative research. The research technique in th...
Motivated by their importance and potential for applications in certain problems in number theory, combinatorics, classical and numerical analysis, and other field of applied mathematics , a variety of polynomials and numbers with their variants and extensions have recently been introduced and investigated. In this sequel, we modify the known gener...
Toric t -designs, or equivalently t -designs on the diagonal subgroup of the unitary group, are sets of points on the torus over which sums reproduce integrals of degree t monomials over the full torus. Motivated by the projective structure of quantum mechanics, we develop the notion of t -designs on the projective torus, which have a much more res...
The Massimult project aims to design and implement an innovative CPU architecture based on combinator reduction with a novel combinator base and a new abstract machine. The evaluation of programs within this architecture is inherently highly parallel and localized, allowing for faster computation, reduced energy consumption, improved scalability, e...
We study the connection between Conway-Coxeter frieze patterns and the data of the minimal resolution of a complex curve singularity: using Popescu-Pampu's notion of the lotus of a singularity, we describe a bijection between the dual resolution graphs of Newton non-degenerate plane curve singularities and Conway-Coxeter friezes. We use representat...
A Computação Bioinspirada é uma área de pesquisa focada no desenvolvimento de técnicas inspiradas em fenômenos da natureza para a solução de problemas de otimização computacional, classificados como problemas NP. Neste trabalho, oAE (Algoritmo Evolutivo) é implementado para a solução do Problema do Percurso do Cavalo (PPC). Este algoritmo é baseado...
Chow rings of flag varieties have bases of Schubert cycles $\sigma _u $, indexed by permutations. A major problem of algebraic combinatorics is to give a positive combinatorial formula for the structure constants of this basis. The celebrated Littlewood–Richardson rules solve this problem for special products $\sigma _u \cdot \sigma _v$, where u an...
The development of the digital era in the 21st century has brought significant changes in various aspects of human life, especially education. Technologies such as artificial intelligence (AI) have been applied in various fields, including education, to improve the efficiency and effectiveness of learning. ChatGPT, one of the text-based AI models,...
The article identifies potential problem areas in the field of probability theory and combinatorics, proposes action algorithms with the active use of step-by-step instructions and diagrams. The study determines the need to study examples with visualization of the structure of the choice of formulas for calculating the desired parameters. The abili...
Consider two simple graphs, G1 and G2, with their respective vertex sets V(G1) and V(G2). The Kronecker product forms a new graph with a vertex set V(G1) X V(G2). In this new graph, two vertices, (x, y) and (u, v), are adjacent if and only if xu is an edge in G1 and yv is an edge in G2. While the adjacency spectrum of this product is known, the dis...
OBJETIVO: mostrar a los investigadores el análisis estadístico de un experimento bifactorial empleando software. MÉTODO: en la descripción, se emplean datos de rendimiento de avena, cultivada en tres fechas de siembra y cuatro niveles de nitrógeno, en un diseño de bloques completos al azar y arreglo combinatorio. Se realizó el análisis de la varian...
Infinite generalizations of theorems in finite combinatorics were initiated by Erdős due to his famous Erdős–Menger conjecture (now known as the Aharoni–Berger theorem) that extends Menger's theorem to infinite graphs in a structural way. We prove a generalization of this manner of the classical result about packing edge‐disjoint T$ T$‐paths in an...
In this note we focus on combinatorial aspects of plus-one generated line arrangements. We provide combinatorial constraints on such arrangements and we present new examples of plus-one generated arrangements constructed by using classical Klein and Wiman reflection arrangements, and we detect, among all known sporadic simplicial arrangements up to...
Resumen. Exploramos un enfoque combinatorio basado en el Triángulo de Pascal para abordar el problema clásico de las sumas de potencias de números naturales. Este enfoque se fundamenta en tres secuencias clave derivadas del triángulo: los coeficientes binomiales, los números de Narayana y una variante de los coeficientes binomiales. A través de est...
We study two-dimensional topological gauge theories with gauge group equal to the symmetric group Sn and their string theory duals. The simplest such theory is the topological quantum field theory of principal Sn fiber bundles. Its correlators are equal to Hurwitz numbers. The operator products in the gauge theory for each finite value of n are cod...
Integral geometry uses four geometric invariants—the Minkowski functionals—to characterize certain subsets of three-dimensional (3D) space. The question was, how is the fluid flow in a 3D porous system related to these invariants? In this work, we systematically study the dependency of permeability on the geometrical characteristics of two categori...
![Figure 1: Partial latin squares with no Sudoku completion [2].](profile/Hollis-Williams/publication/386106991/figure/fig1/AS:11431281292473775@1732575102165/Partial-latin-squares-with-no-Sudoku-completion-2_Q320.jpg)
![Figure 2: Example of a completed Sudoku latin square of type (3, 3) [3].](profile/Hollis-Williams/publication/386106991/figure/fig2/AS:11431281292421724@1732575102422/Example-of-a-completed-Sudoku-latin-square-of-type-3-3-3_Q320.jpg)
![Figure 3: A solved Pentadoku puzzle [2].](profile/Hollis-Williams/publication/386106991/figure/fig3/AS:11431281292473776@1732575102618/A-solved-Pentadoku-puzzle-2_Q320.jpg)
Sudoku is a well-known recreational puzzle which has been studied from the perspectives of group theory and algebraic combinatorics. In this article, we consider applications of Sudoku puzzle constructions in several areas of pure and applied mathematics, including some deep results related to tilings.
We review the relationship between discrete groups of symmetries of Euclidean three-space, constructions in algebraic geometry around Kleinian singularities including versions of Hilbert and Quot schemes, and their relationship to finite-dimensional and affine Lie algebras via the McKay correspondence. We focus on combinatorial aspects, such as the...
To better handle real-world uncertainty, concepts such as fuzzy sets, neutrosophic sets, rough sets, and soft sets have been introduced. For example, neutrosophic sets, which simultaneously represent truth, indeterminacy, and falsehood, have proven to be valuable tools for modeling uncertainty in complex systems. These set concepts are increasingly...
Many prominent combinatorial sequences, such as the Fibonacci, Lucas, Pell, Jacobsthal and Tribonacci sequences, are defined by homogeneous linear recurrence relations with constant coefficients. These sequences are often referred to as C-finite sequences, and a variety of representations have been employed throughout the literature, largely influe...
This study contributes to previous research on collaborative approaches to instruction in mathematics. The study focuses on the relationships between the type of talk in groups and individual students' combinatorial thinking. Four case studies of mixed-attainment groups of middle-school students working with mathematical problem solving in combinat...
We give an explicit description of two operad structures on the species composition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{p}\circ \textbf{q}$$\end{doc...