Combinatorics and Algorithms Design

Learn fundamental concepts and advanced techniques in combinatorial mathematics and algorithm design through this comprehensive course from Tsinghua University. Master key topics including magic squares, password security, permutations, combinations, generating functions, recurrence relations, Fibonacci sequences, Catalan numbers, and Stirling numbers. Explore essential algorithm design paradigms such as incremental algorithms, divide-and-conquer strategies, randomized algorithms, and dynamic programming. Apply mathematical principles to solve practical problems involving the inclusion-exclusion principle, pigeonhole principle, and algorithm complexity analysis. Develop skills in asymptotic analysis, loop invariants, and various algorithmic methods while working through hands-on examples like the majority element problem, Dutch national flag problem, merge sort, and knapsack optimization. Conclude with advanced topics in group theory, including Burnside's lemma, Polya's theorem, and applications to rotating polyhedrons, with regular homework assignments and demonstrations reinforcing learning throughout the course.