Some combinatorial problems

AbstractThere are many interesting and sophisticated problems posed in the IMO, Putnam and domestic Olympiads. Some of these problems have deep mathematical background, nice generalizations, and lead to new areas of research in combinatorics. We investigate several topics in this category and mention some results and open problems

Some Combinatorial Problems on Binary Matrices in Programming Courses

The study proves the existence of an algorithm to receive all elements of a class of binary matrices without obtaining redundant elements, e. g. without obtaining binary matrices that do not belong to the class. This makes it possible to avoid checking whether each of the objects received possesses the necessary properties. This significantly improves the efficiency of the algorithm in terms of the criterion of time. Certain useful educational effects related to the analysis of such problems in programming classes are also pointed out

Combinatorial problems in finite geometry and lacunary polynomials

We describe some combinatorial problems in finite projective planes and indicate how R\'edei's theory of lacunary polynomials can be applied to them

Cluster algebras and representation theory

We apply the new theory of cluster algebras of Fomin and Zelevinsky to study some combinatorial problems arising in Lie theory. This is joint work with Geiss and Schr\"oer (3, 4, 5, 6), and with Hernandez (8, 9)

Combinatorial problems in finite fields and Sidon sets

We use Sidon sets to present an elementary method to study some combinatorial problems in finite fields, such as sum product estimates, solubility of some equations and distribution of sequences in small intervals. We obtain classic and more recent results avoiding the use of exponential sums, the usual tool to deal with these problems.Comment: 13 page

Rational method of generation of combinations for parallel calculations in some combinatorial problems

Целью данной статьи являлась разработка нового способа нумерации сочетаний. Его особенностью является отсутствие в алгоритме циклов и ветвлений, что позволяет эффективно использовать его в многопоточном режиме. Для выполнения поставленной задачи были произведены разработка алгоритма, нахождение оптимального способа вычисления требуемых величин и оптимизация под многопоточные системы. По сравнению с классическими, данный метод показывает заметное увеличение производительности даже не в самых благоприятных условиях. Особенностью полученного алгоритма является независимое выполнение разных потоков вычисления, что очень важно при выполнении программы на процессорах SIMD архитектуры. Таким образом, разработанный способ нумерации имеет очевидные преимущества и может быть использован в задачах, решаемых методами комбинаторной оптимизации.The purpose of this paper is development of a new method of numbering combinations. Its feature is the absence of loops and branches in the algorithm, which allows using it effectively multithreaded. For this purpose the following tasks have been produced: development of an algorithm, finding an optimal method for calculating the required values and optimization for multi-threaded system. Compared to classical ones, proposed method showed a significant performance increase even in adverse conditions. Thus, the developed algorithm of numbering has obvious advantages, and can be used in tasks, solved by methods of combinatorial optimization