PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Combinatorics ↓


Enumerative Combinatorics


Enumerative combinatorics is a branch of mathematics concerned with counting the number of ways in which certain patterns can be formed. It attempts to answer questions such as "In how many different ways can a particular structure be arranged?" or "How many possible configurations exist under a given set of restrictions?" This field is fundamental to many areas of mathematics as it provides the essential foundation for probability, algorithm analysis, and other domains.

Basic principles of counting

The fundamental principle of counting, sometimes called the law of product, is important in combinatorics. It states that if you have a sequence of options to choose from, and there are n ways to choose the first option and m ways to choose the second option, then there are n × m ways to form the sequence of options. This principle can be generalized to any number of options.

Example

Imagine you are choosing an outfit. You have 3 shirts (red, blue, yellow) and 2 pants (jeans, shorts). You want to know how many different outfits can be made. Using the product rule, you calculate:

Number of outfits = Number of shirts × Number of pants = 3 × 2 = 6

Possible outfit combinations include: (red, jeans), (red, shorts), (blue, jeans), (blue, shorts), (yellow, jeans), (yellow, shorts).

Permutation

Permutation refers to arranging objects in a sequence where the order matters. The number of permutations of n distinct objects is given by the factorial of n, denoted as n!.

Formula

n! = n × (n-1) × (n-2) × ... × 2 × 1

Example

If we have to watch 4 movies on consecutive nights, how many viewing sequences are possible?

Number of permutations = 4! = 4 × 3 × 2 × 1 = 24

Thus, there are 24 different possible orders of watching these four movies.

Visual example

item 1 Item 2 Item 3 Item 4 There are 4 total permutations! : 24 possible arrangements

Combination

Combination refers to the selection of items regardless of the order. If we have a set of items and we want to select some of them, the number of ways to do so is called a combination. The number of combinations of n items taken r at a time is represented by C(n, r) or nCr, and can be calculated using the formula:

Formula

C(n, r) = n! / (r! × (nr)!)

Example

If you have 5 different books, but you can only take 2 on vacation, the number of combinations is determined by:

C(5, 2) = 5! / (2! × (5-2)!) = 10

This means that you can choose from 10 different pairs of books.

Visual example

1 2 3 4 Picking pairs: C(4,2) = 6 options

Binomial theorem and Pascal's triangle

The binomial theorem gives a formula for expanding expressions that are raised to exponents. For any integer n, it is represented as:

Formula

(x + y)^n = ∑(C(n, k) × x^(nk) × y^k), where 0 ≤ k ≤ n

Pascal's triangle is a wonderful tool for finding the coefficients of terms in expansions. The nth row of Pascal's triangle consists of the numbers C(n,0), C(n,1), ..., C(n, n).

Example

(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

Visual example

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

Inclusion-exclusion principle

The inclusion-exclusion principle is a fundamental but sometimes counter-intuitive principle in combinatorics, used to count the number of elements in the union of several sets that can intersect. For two sets A and B, the principle is expressed as follows:

Formula

|A ∪ B| = |A| + |B| - |A ∩ B|

This principle can be applied to multiple sets as well.

Example

Suppose a university offers 3 types of courses: science (60 students), math (50 students), and both (15 students). The total number of students enrolled in either science or math classes is:

|Science ∪ Math| = |Science| + |Math| - |Science ∩ Math| = 60 + 50 - 15 = 95

Visual example

A B A ∩ B

Generating function

Generating functions are a powerful tool in combinatorics when dealing with sequences. They turn combinatorial problems into algebraic problems and make it easier to recognize patterns and find closed formulas. The generating function for a sequence (a n ) is represented as follows:

Formula

G(x) = a 0 + a 1 x + a 2 x² + a 3 x³ + ...

Example

Consider a sequence of numbers indicating the number of subsets of a set with n elements:

The generating function is:

G(x) = (1 + x)^n

Applications of computational combinatorics

Enumerative combinatorics has a wide range of applications in fields such as computer science (to analyze algorithms), mathematics (especially probability theory and graph theory) and statistics. By understanding the configuration and arrangement of various sets and structures, enumerative combinatorics helps solve complex counting problems efficiently.

Examples in computer science

In computer science, the analysis of the complexity of algorithms often relies on combinatorics. For example, sorting algorithms can benefit greatly from combinatorics analysis to determine the best, average, and worst scenarios.

Examples in probability

In probability theory, the concepts of permutation and combination are used to calculate the probability of various events occurring in a structured sample space.

By understanding these core topics and their visualization, you have taken a valuable step into the field of computational combinatorics. This field opens many doors to a deeper understanding of both theoretical and applied mathematics.


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Enumerative Combinatorics

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