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Unformatted text preview: MATH 681 Notes Combinatorics and Graph Theory I 1 What is combinatorics? Combinatorics is the branch of mathematics dealing with things that are discrete , such as the integers, or words created from an alphabet. This is in contrast to analysis, which deals with the properties of continuous systems, such as the real number line or a differential equation. Many (but not all) combinatorial problems also address systems which are finite , so the systems being investigated will have a specific number of elements: in fact, the question how many elements are in a particular system? is a common combinatorial query. The above loose definition of combinatorics is inclusive of a few other disciplines. Explorations of the arithmetic properties of the integers may be considered to fall under the heading of number the- ory ; explorations of multiplication-like and addition-like operations on discrete structures generally falls under the heading of algebra . The distinctions between these disciplines and combinatorics, especially as regards finite structures, are somewhat amorphous. Broadly speaking, combinatorics tends to ask four types of questions: Enumerative questions How large is a particular set? Existential questions Is a set of conditions on a discrete system actually satisfiable? Extremal questions How large does a discrete structure need to become before certain substruc- tures become inevitable? Computational questions How long would it take a computer to answer a question posed about a discrete system? To give some context to the above descriptions, here are a few fundamentally combinatorial ques- tions: How many numbers less than 10000 consist of distinct digits appearing in ascending order? How many people need to be attending a party in order to guarantee that there are either 5 mutual acquaintances or 5 mutual strangers? If 20 people each choose a set of acceptable roommates, can each person be paired with an acceptable roommate? How could a computer program be written to find such a pairing? How many anagrams are there of the word MISSISSIPPI? If we remove one square from a 7 9 checkerboard, can the remaining squares be covered with dominoes? Does it matter which square we remove? If we walk five blocks north and six blocks east, taking a random route, what is the probability that we will pass by a newsstand one block to the north and one block east of our starting position? After we have completed both semester of this course, you will know the answer to five of these questions as well as many others! Page 1 of 10 August 25, 2009 MATH 681 Notes Combinatorics and Graph Theory I 2 Introductory Enumerative Methods The absolute simplest enumerative method is exhaustive enumeration , also known as brute force: simply list out every Question 1: How many three-digit numbers are there whose digits add up to 6?...
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