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Lecturer(s)
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Course content
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1. General proof principles: mathematical induction, the sum, product, complement, and bijection principles. 2. Basic concepts of combinatorics: variations, permutations, combinations. 3. Combinatorial identities: Pascal?s triangle, the binomial theorem and its consequences, various identities involving binomial coefficients. 4. Lattice paths: deriving identities via path counting, the reflection principle, Catalan numbers and their significance in combinatorics. 5. Stirling numbers of the second kind and related enumerative problems. 6. Permutations: the relationship between permutations and symmetries of geometric figures, decomposition of permutations into transpositions, the sign of a permutation and its properties, applications in mathematical puzzles. 7. The Pigeonhole Principle and its application in various types of problems. 8. Principle of inclusion and exclusion: counting permutations without fixed points, the number of surjective mappings, Euler?s totient function. 9. Combinatorics of partitions: an introduction to the theory of integer partitions and fundamental results.
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Learning activities and teaching methods
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unspecified
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Learning outcomes
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The course aims to introduce students to the basic methods and principles of combinatorics and to develop their ability to accurately count and analyze discrete structures.
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Prerequisites
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unspecified
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Assessment methods and criteria
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unspecified
- Attendance at exercises and submission of assigned homework. - Successful completion of two tests.
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Recommended literature
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Bóna M. (2017). A walk through combinatorics: an introduction to enumeration and graph theory. World Scientific.
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HERMAN, J., KUČERA, R., ŠIMŠA, J. (1997). Metody řešení matematických úloh II. Brno.
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MATOUŠEK, J., NEŠETŘIL, J. (2009). Invitation to Discrete Mathematics. Oxford University Press.
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Matoušek, J., Nešetřil, J. (2009). Kapitoly z diskrétní matematiky. Praha: Karolinum.
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Švrček J. (2003). Úvod do kombinatoriky. VUP OLomouc.
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