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Introduction to Combinatorics

The combinatorics is a category of mathematics regarding the study of finite or countable discrete structures. The feature of combinatorics involve counting the structures of a given type and size, deciding whenever the definite criteria met, and investigating and constructing objects meeting the criteria, finding minimum, maximum, or optimal objects and studying the combinatorial structures occurring in an algebraic context or applying algebraic techniques to combinatorial problems.

The combinatorial problems occur in numerous regions of pure mathematics, extraordinarily in the region of probability theory, algebra, topology, and geometry and combinatorics also has many applications in the region of optimization, computer science, ergodic theory and in statistical physics. Different combinatorial questions have historically been considered in isolation giving an ad hoc result to a problem occurring in some mathematical context. In late 20th century, however, powerful and common theoretical techniques were developed, and make combinatorics into a self-governing branch of mathematics in its own right. The oldest and most accessible section of combinatorics is graph theory that also has numerous natural connections to other regions. Combinatorics is commonly used in computer science to acquire formulas and estimate the analysis of algorithms.

** **The approaches and subfields of combinatorics:

**Enumerative combinatorics:**

The enumerative combinatorics focuses on counting the number of certain combinatorial objects. It is the most standard region of combinatorics, and. however counting the number of elements in a set is instead broad mathematical problem, and various problems which arise in applications have a relatively simple combinatorial explanation. Fibonacci number is the fundamental instance of a problem in enumerative combinatorics. The twelvefold manner provides a unified framework for counting the partitions, permutations and combinations.

Analytic combinatorics:

Analytic combinatorics focuses on the enumeration of combinatorial structures by using tools from probability theory and complex analysis. In distinction with enumerative combinatorics that uses explicit combinatorial formulae and generating functions to define the answers, analytic combinatorics aims at acquiring the asymptotic formula.

**Extremal combinatorics:**

Extremal combinatorics studies the extremal questions on set systems. The questions addressed in this situation are about the largest possible graph that satisfies some properties. For illustration- the largest triangle-free graph on 2n vertices is a complete bipartite graph Kn, n. At times it is too hard even to find the extremal answer f (n) exactly and one can only give an asymptotic estimate.** **

**Probabilistic combinatorics:**

In probabilistic combinatorics, the questions are of the kind as follows: what is the probability of certain property for an arbitrary discrete object, such as the random graph? For illustration, what is the average number of triangles in an arbitrary graph? The Probabilistic techniques are also used to determine the existence of the combinatorial objects with certain prescribed properties, simply by observing the probability of arbitrarily selecting an object with those properties which is greater than 0. This approach termed to as the probabilistic technique proved highly efficient in applications to the extremal combinatorics and graph theory.

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