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Mathematics via Problems: Part 3: Combinatorics
* 3: Combinatorics.
Providence :
AMS, 2023.
Compiler: Skopenkov M., A. Zaslavsky
This book is a translation from Russian of Part III of the book Mathematics via Problems: From Olympiads and Math Circles to Profession. Part I, Algebra, and Part II, Geometry, have been published in the same series.
The main goal of this book is to develop important parts of mathematics through problems. The authors tried to put together sequences of problems that allow high school students (and some undergraduates) with strong interest in mathematics to discover such topics in combinatorics as counting, graphs, constructions and invariants in combinatorics, games and algorithms, probabilistic aspects of combinatorics, and combinatorial geometry.
Definitions and/or references for material that is not standard in the school curriculum are included. To help students that might be unfamiliar with new material, problems are carefully arranged to provide gradual introduction into each subject. Problems are often accompanied by hints and/or complete solutions.
The book is based on classes taught by the authors at different times at the Independent University of Moscow, at a number of Moscow schools and math circles, and at various summer schools. It can be used by high school students and undergraduates, their teachers, and organizers of summer camps and math circles.
In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, SLMath (formerly MSRI) and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.
Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).
Readership
High school students and their mentors and teachers.
Chapters
Arzhantsev I., , in: Mathematics via Problems: Part 3: Combinatorics* 3: Combinatorics.: Providence: AMS, 2023. Ch. 7.5 P. 161–166.
Added: August 27, 2024
Skopenkov M., Ustinov A., Zaslavsky A., , in: Mathematics via Problems: Part 3: Combinatorics* 3: Combinatorics.: Providence: AMS, 2023.
Added: October 16, 2025
Zlotnik A., Математические заметки 2026 Т. 120 № 6 С. 1005–1009
Численным методам решения систем газодинамических уравнений посвящена обширная литература. Ранее было разработано и успешно апробировано специальное семейство симметричных по пространству консервативных разностных методов, основанных на предварительной кинетической, точнее, квазигазодинамической (КГД), регуляризации этих уравнений. Актуальной задачей является построение численных методов, которые обладают не только свойством консервативности по массе, импульсу и полной энергии, но и удовлетворяют условиям энтропийной ...
Added: October 1, 2026
Vyugin I. V., Sashadhar D., Algebra and Number Theory 2026 P. 1–10
We study the K-Fibonacci sequence Fp modulo prime p. Cardinalities of sets |Fp+Fp| and |Fp⋅Fp| are estimated. We present the method of estimating doubling constant of some m-dimensional recurrent sets in Fp. ...
Added: October 1, 2026
Bernardin C., João P. M., Gonçalves P., PROBABILITY AND MATHEMATICAL PHYSICS 2026 P. 59–59
We investigate a boundary-driven Ginzburg-Landau dynamics with long-range interactions. In the hydrodynamic limit, the macroscopic evolution is governed by a fractional heat equation with Dirichlet boundary conditions, while the corresponding stationary profile is characterized by a fractional Laplace equation. We establish a dynamical large deviations principle for the empirical measure and derive the associated stationary ...
Added: October 1, 2026
Kuksin S., Dynamical Systems 2026
We study the mixing properties of discrete-time and continuous-time dissipative dynamical systems driven by bounded mixing random forces. The continuous-time systems are
reduced to discrete-time random dynamical systems generated by time-one maps, so that
the main analysis is carried out in the discrete setting. We introduce a class of mixing random forcings whose regular conditional distributions with ...
Added: October 1, 2026
Kuksin S., Shirikyan A., Journal of Dynamics and Differential Equations 2026 P. 1098–1100
The paper deals with the problem of large-time behaviour of trajectories for discrete-time dynamical systems driven by a random noise. Assuming that the phase space is finite-dimensional and compact, and the noise is a Markov process with a transition probability satisfying some regularity hypotheses, we prove that all the trajectories converge to a unique measure ...
Added: October 1, 2026
Potanin B., Dolgikh S., Statistics and Probability Letters 2026 Article 110984
We derive bounds on the gradient and Hessian of the log-CDF, ln F(x), of the multivariate normal distribution. These bounds scale linearly and quadratically in ‖x‖ , respectively, with constants depending only on the covariance matrix. We demonstrate the usefulness of these bounds by proving asymptotic normality of the maximum-likelihood estimator of the multivariate probit ...
Added: October 1, 2026
A. V. Pereskokov, Journal of Mathematical Sciences 2026 Vol. 302 No. 4 P. 531–545
We consider the Zeeman effect problem for the hydrogen atom in a magnetic field using
irreducible representations of the Karasev–Novikova algebra with quadratic commutation
relations. We find the asymptotics of a series of eigenvalues and the corresponding
asymptotic eigenfunctions near the upper boundaries of spectral clusters. ...
Added: October 1, 2026
Yakovlev K., Puchkin N., Journal of Complexity 2026 Vol. 97
We present a theory for simultaneous approximation of the score function and its derivatives, enabling the handling of data distributions with low-dimensional structure and unbounded support. Our approximation error bounds match those in the literature while relying on assumptions that relax the usual bounded support requirement. Crucially, our bounds are free from the curse of ...
Added: September 30, 2026
Zlotnik A., Mathematical notes 2026 Vol. 120 No. 6 P. 1174–1178
Numerical methods for solving systems of gas dynamic equations are the subject of a vast literature. A special family of spatially symmetric conservative difference methods based on preliminary kinetic, or quasi-gasdynamic (QGD), regularization of these equations was constructed and successfully tested. A pressing issue is the construction of numerical methods that are not only conservative ...
Added: September 30, 2026
Beldiev I., Тимашёв Д. А., Алгебра и анализ 2026 Т. 38 № 5 С. 1–10
An algebraic variety X is called a homogeneous space if there exists a transitive regular action of an algebraic group on X. We prove inequalities between the dimension of a homogeneous space of a linear algebraic group and its Picard number. ...
Added: September 30, 2026
Planche L., Ilina A., Jay F. et al., Molecular Biology and Evolution 2026 Article 235
Admixture between populations is a common feature of human history. Admixture events introduce new genetic variation that can fuel evolution. Characterizing the significance of admixture events on the evolution of populations across various species is of great interest to evolutionary geneticists. Local Ancestry Inference (LAI) methods infer genetic ancestry of an individual at a particular ...
Added: September 30, 2026
Glutsyuk A., Ilyashenko Y., Izvestiya. Mathematics 2026 Vol. 90 No. 1 P. 73–89
We show that a completely non-degenerate B-group is uniquely determined by its factor: two such groups with conformally equivalent factors are Möbius conjugate. A similar property is inherent to the quasi-Fuchsian groups but not to degenerate B-groups. We also study the factor of a B-group as a triple: the main factor, the marked characteristic complex, and a homotopy class of ...
Added: September 30, 2026
Tuzhilin M., Автоматика и телемеханика 2026 № 11 С. 84–97
Предлагается обобщение двух известных инвариантов реальных сетей: степени и кси-центральности. Строится серия центральностей, основанная на матрице Лапласа сети и параметризованная параметром j со следующими свойствами: во-первых, при j = 0, 1 эти центральности совпадают со степенью и ксицентральностью; во-вторых, их распределение хорошо приближается распределением Вейбулла; в-третьих, для реальных сетей они имеют правостороннюю асимметрию, а для ...
Added: September 29, 2026
Melnikov I., Pelinovsky E., Доклады Российской академии наук. Физика, технические науки (ранее - Доклады Академии Наук. Физика) 2026 Т. 529 С. 29–36
Conditions for the emergence of pyramidal solitary waves (solitary waves possessing more than two inflection points) are presented for a family of generalized Korteweg – de Vries (KdV) equations. For the generalized Gardner equation, it is shown that pyramidal solitons are unstable. The decay of initial perturbations close to pyramidal solitary waves is demonstrated numerically. ...
Added: September 29, 2026
Merkulov S., Journal of Pure and Applied Algebra 2026 Vol. 230 P. 1–19
We study the dual cyclic Hochschild complex $Cyc(A,\K)$ of a
(possibly, infinite-dimensional) $A_\infty$-algebra $(A,\mu)$ and prove
that any pre-Calabi-Yau extension $\pi$ of the given $A_\infty$ structure $\mu$ in $A$
induces on the cyclic cohomology of $(A,\mu)$ a representation of a new dg properad of {\em oriented}\, ribbon graphs. We compute the cohomology of that properad in terms of ...
Added: September 29, 2026
Хрыстик М. А., European Journal of Combinatorics 2026 Vol. 136 P. 104393–104393
Let 𝑓𝑊(𝑛) be the number of different factors of length 𝑛 appearing in a word 𝑊. In a classical result by Morse and Hedlund, a characterization is given for infinite words satisfying the condition 𝑓𝑊(𝑛)≤𝑛 for some 𝑛∈ℕ. In this paper, we describe the form of finite words that satisfy the condition 𝑓𝑊(𝑛)≤𝑛. We study ...
Added: September 28, 2026
Teregulov T., Loubenets E. R., / Series Quantum Physics "arXiv". 2026. No. 2609.31472.
We develop a new three-party quantum secret-sharing (QSS) protocol based on a three-qubit W state. This protocol encodes the secret-sequence bits using X gates and employs randomly selected Hadamard operations and measurement bases to generate information and security-test rounds. We evaluate the efficiency of the proposed protocol and analyze its security against an internal adversary ...
Added: September 28, 2026
Здержиков Н. И., Shabanov D. A., Степанов И. Д., Чебышевский сборник 2026 Т. 27 № 3 С. 74–90
In this article we study one of many possible generalizations of the notion of a proper coloring when applied to hypergraphs. Namely, the so called fractional \((a:b)\)-colorings are dealt with. Such coloring is a way to match each vertex of the hypergraph with exactly \(b\) colors (out of \(a\) available ones), in a manner that ...
Added: September 28, 2026
Guterman A., Jonoska N., Kreines E. et al., Proceedings of the Edinburgh Mathematical Society 2026 Vol. 69 No. 3 P. 1041–1057
We provide four equivalent combinatorial conditions for a simple assembly graph (rigid vertex graph where all vertices are of degree 1 or 4) to have the largest number of Hamiltonian sets of polygonal paths relative to its size. These conditions serve to prove the conjecture that such a maximum, which is equal to 𝐹_(2𝑛+1) −1, ...
Added: September 27, 2026
Shishkina E., Современная математика. Фундаментальные направления 2026 Т. 72 № 1 С. 52–67
In this paper, we construct a weighted Sobolev space of fractional order based on the
generalized Bessel potential.We apply these results to the analysis of the singular fractional Schr¨odinger
equation. To solve the Cauchy problem for this equation, we prove an estimate that relates the norm of
the solution to the norm of the initial condition in the ...
Added: September 26, 2026
Shishkina E., Computational Mathematics and Mathematical Physics 2026 Vol. 66 No. 5 P. 804–815
This article demonstrates that the Laplace–Bessel operator generates a strongly continuous
semigroup on a weighted Lebesgue space. Using this semigroup, we define the fractional Laplace–
Bessel operator via Balakrishnan’s formula. Furthermore, we derive three distinct representations for
fractional powers of the negative Laplace–Bessel operator. ...
Added: September 26, 2026
Springer, 2025.
The second edition of this 5-volume handbook is intended to be a basic yet comprehensive reference work in combinatorial optimization that will benefit newcomers and researchers for years to come. This multi-volume work deals with several algorithmic approaches for discrete problems as well as with many combinatorial problems. The editors have brought together almost every aspect ...
Added: January 18, 2026
Bille A., Buchstaber V., Coste S. et al., Journal of Physics A: Mathematical and Theoretical 2025 Vol. 58 No. 2 Article 025212
We analyze the numbers of closed paths of length 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, we refer to such ...
Added: August 22, 2025
Khestanov R., Suvalko A., Социологическое обозрение 2025 Т. 24 № 2 С. 164–189
The article examines the genesis of the concept of novelty and argues that modern difficulties in defining and identifying the new are a symptom of a semantic crisis, which is based on the loss of strategic normativity of novelty in modern culture. Three historical-semantic models of novelty are identified and analyzed: cyclical (characteristic of prehistoric ...
Added: June 29, 2025