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Mathematics
Combinatorics
1. Introduction to Combinatorics
2. Fundamental Counting Principles
3. Permutations
4. Combinations
5. The Binomial Theorem
6. Advanced Counting Techniques
7. Recurrence Relations
8. Generating Functions
9. Special Counting Numbers and Sequences
10. Graph Theory and Combinatorics
11. Design Theory
12. Probabilistic Combinatorics
13. Algebraic Combinatorics
14. Extremal Combinatorics
15. Applications and Advanced Topics
11.
Design Theory
11.1.
Block Designs
11.1.1.
Definition and Parameters
11.1.2.
Incidence Structures
11.1.3.
Balanced Incomplete Block Designs
11.1.3.1.
Definition and Properties
11.1.3.2.
Necessary Conditions
11.1.3.3.
Fisher's Inequality
11.1.4.
Symmetric Designs
11.1.5.
Resolvable Designs
11.2.
Steiner Systems
11.2.1.
Definition and Notation
11.2.2.
Steiner Triple Systems
11.2.3.
Steiner Systems S(t,k,n)
11.2.4.
Existence Conditions
11.2.5.
Construction Methods
11.3.
Latin Squares
11.3.1.
Definition and Properties
11.3.2.
Order of Latin Squares
11.3.3.
Orthogonal Latin Squares
11.3.4.
Mutually Orthogonal Latin Squares
11.3.5.
Applications in Experimental Design
11.3.6.
Euler's 36 Officers Problem
11.4.
Hadamard Matrices
11.4.1.
Definition and Properties
11.4.2.
Hadamard's Conjecture
11.4.3.
Construction Methods
11.4.4.
Applications to Coding Theory
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12. Probabilistic Combinatorics