Principles and Techniques in Combinatorics

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Aspects of combinatorics include: counting the structures of a given kind and size, deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria. Several useful combinatorial rules or combinatorial principles are commonly recognized and used. Each of these principles is used for a specific purpose. The rule of sum addition rule , rule of product multiplication rule , and inclusion-exclusion principle are often used for enumerative purposes.

Bijective proofs are utilized to demonstrate that two sets have the same number of elements.

Principles and Techniques in Combinatorics Chen Chuan Chong Koh Khee Meng WS

Double counting is a method of showing that two expressions are equal. The pigeonhole principle often ascertains the existence of something or is used to determine the minimum or maximum number of something in a discrete context.

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Generating functions and recurrence relations are powerful tools that can be used to manipulate sequences, and can describe if not resolve many combinatorial situations. Each of these techniques is described in greater detail below. More formally, the sum of the sizes of two disjoint sets is equal to the size of the union of these sets.

The inclusion-exclusion principle is a counting technique that is used to obtain the number of elements in a union of multiple sets. This counting method ensures that elements that are present in more than one set in the union are not counted more than once.

Principles And Techniques In Combinatorics

It considers the size of each set and the size of the intersections of the sets. See the diagram below for an example with three sets. A bijective function is one in which there is a one-to-one correspondence between the elements of two sets. Double counting is a combinatorial proof technique for showing that two expressions are equal. This is done by demonstrating that the two expressions are two different ways of counting the size of one set. Since both expressions equal the size of the same set, they equal each other.

This principle allows one to demonstrate the existence of some element in a set with some specific properties. For example, consider a set of three gloves. In such a set, there must be either two left gloves or two right gloves or three of left or right.

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This is an application of the pigeonhole principle that yields information about the properties of the gloves in the set. Generating functions can be thought of as polynomials with infinitely many terms whose coefficients correspond to the terms of a sequence. A recurrence relation defines each term of a sequence in terms of the preceding terms. In other words, once one or more initial terms are given, each of the following terms of the sequence is a function of the preceding terms.

The Fibonacci sequence is one example of a recurrence relation. Thus, the sequence of Fibonacci numbers begins:. A permutation of a set of objects is an arrangement of those objects in a particular order; the number of permutations can be counted. A permutation of a set of objects is an arrangement of those objects into a particular order.

One might define an anagram of a word as a permutation of its letters. The 6 permutations of 3 balls: If one has three different colored balls, there are six distinct ways to order them, as shown. These six distinct orderings are as follows: red-green-blue, red-blue-green, green-red-blue, green-blue-red, blue-red-green, and blue-green-red.

So [latex]5! In the game of Solitaire, seven cards are dealt out at the beginning: one face-up, and the other six face-down. What makes this a permutation problem is that the order matters: if an ace is hiding somewhere in those six cards, it makes a difference whether the ace is on the first position, the second, etc. Permutation problems can always be addressed as an example of the multiplication rule, with one small twist.

One stack of cards in a game of solitaire: To find out how many possible combinations of cards there are below the seven of spades, we use the concept of permutations to calculate the possible arrangements of cards. The answer is If any given card is in the first position, how many cards might be in second position?

The seven of spades and the next card have both been dealt. So there are possible cards left for the second position. So how many possibilities are there for the first two positions combined? How many possibilities are there for all six positions? Instead of typing into a calculator six numbers to multiply, or sixty numbers or six hundred depending on the problem, the answer to a permutation problem can be found by dividing two factorials.

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting rearranging objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order.

The Fundamental Counting Principle

The study of permutations generally belongs to the field of combinatorics. Permutations occur, in more or less prominent ways, in almost every domain of mathematics.

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They often arise when different orderings on certain finite sets are considered, possibly only because one wants to ignore such orderings and needs to know how many configurations are thus identified. For similar reasons, permutations arise in the study of sorting algorithms in computer science. The number of permutations of distinct elements can be calculated when not all elements from a given set are used.

Recall that, if all objects in a set are distinct, then they can be arranged in [latex]n!

The quantity [latex]n! It is easy enough to use this formula to count the number of possible permutations of a set of distinct objects; for example, the number of permutations of three differently-colored balls. However, consider a situation where not all of the elements in a set of distinct objects are used in each permutation. In this case, not all of the cards from the deck are chosen for each possible permutation.

There exists a formula for solving permutation problems such as this one, which would otherwise be nearly impossible to determine. If not all of the objects in a set of unique elements are chosen, the following formula is used. Plugging these values into the formula, we have:. Remember that both [latex]25! Circular permutations -- 1. Combinations -- 1. The injection and bijection principles -- 1. Arrangements and selections with repititions -- 1.

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Distribution problems -- Exercise 1 -- 2. Binomial coefficients and multinomial coefficients. Introduction -- 2. The Binomial Theorem -- 2. Combinatorial identities -- 2. The Pascal's Triangle -- 2. Chu Shih-Chieh's Identity -- 2. Shortest routes in a rectangular grid -- 2. Some properties of Binomial coefficients -- 2.

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Multinomial coefficients and the Multinomial Theorem -- Exercise 2 -- 3. The Pigeonhole Principle and Ramsey Numbers. Introduction -- 3. The Pigeonhole Principle -- 3. More examples -- 3. Ramsey type problems and Ramsey numbers -- 3. Bounds for Ramsey numbers -- Exercise 3 -- 4. The Principle of Inclusion and Exclusion. Introduction -- 4. The principle -- 4. A generalization -- 4.