packageSOS - Package for MATH 239 Final Exam Prepared by...

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Package for MATH 239, Final Exam Prepared by Abel Molina 1 August 6, 2011 Contents 1 Enumeration 2 1.1 Basic tools for enumeration . . . . . . . . . . . . . . . . . . . 2 1.2 Generating functions . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Properties of generating functions . . . . . . . . . . . . . . . . 4 1.3.1 The Sum Lemma . . . . . . . . . . . . . . . . . . . . . 4 1.3.2 The Product Lemma . . . . . . . . . . . . . . . . . . . 4 1.4 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . 5 1.5 Recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Solving problems using generating functions . . . . . . . . . . 6 1.6.1 Partitions of an integer . . . . . . . . . . . . . . . . . 7 1.6.2 Binary strings . . . . . . . . . . . . . . . . . . . . . . . 8 1.6.3 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6.4 Binary trees . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6.5 Bivariate generating functions . . . . . . . . . . . . . . 11 1.7 Homogeneous recurrence relations . . . . . . . . . . . . . . . 12 2 Introduction to Graph Theory 13 2.1 Families of graphs . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Graph isomorphism . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Adjacency and incidence matrices . . . . . . . . . . . . . . . . 15 2.4 Paths and cycles . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.7 Breadth-first search . . . . . . . . . . . . . . . . . . . . . . . 20 2.8 Planarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1 [email protected] 1
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2.9 Conditions for planarity . . . . . . . . . . . . . . . . . . . . . 24 2.10 Colourability . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.11 Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.12 Bipartite matching . . . . . . . . . . . . . . . . . . . . . . . . 28 2.13 Edge-colourings . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 Tips for the exam 32 1 Enumeration We will speak about sets whose elements are called configurations, each of them with an associated non-negative weight. In our problems, we will want to know how many configurations are there of an specific weight. 1.1 Basic tools for enumeration Definition 1. For any real number a and non-negative integer k , we define a choose k , denoted ( a k ) , as a ( a - 1) . . . ( a - k + 1) k ( k - 1) . . . 1 . Remark 1. If a is equal to a positive integer n , then ( n k ) is the number of subsets of size k of a set of size n : Proof. Let L be the set of all ordered lists of k distinct numbers from the set of size n . There are n ways to choose the first element, n - 1 ways to choose the second, and so on, so | L | = n ( n - 1) ... ( n - k + 1). However, there are k * ( k - 1) * ( k - 2) . . . * 1 ways to permute each list of k elements (by choosing the first element, then the second element, and so on). Hence, the number of ways to choose subsets of size k from n elements is n ( n - 1) ... ( n - k +1) k ( k - 1) ... 1 = ( n k ) Remark 2. If a is equal to a negative integer - n , then ( a k ) is ( - 1) k ( n + k - 1 n - 1 ) . Also, if a = 1 2 , then ( 1 2 k ) = ( - 1) k - 1 2 2 k - 1 k ( 2 k - 2 k - 1 ) Definition 2 (Bijective function) . A function f : X - Y is bijective if for each y Y there is exactly one x X such that f ( x ) = y . A way to show that two expressions are equivalent is to associate each of them with a set, prove that they are equal to the number of elements in the set, and give a bijection between the sets.
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1.2 Generating functions Generating functions are a way of writing down the answer to a counting problem. If there are 3 configurations of weight 0, 5 configurations of weight 1, 6 configurations of weight 2 and so on, the generating function for the problem is 3 + 5 x + 6 x 2 + . . . More formally: Definition 3. Let S be a set of configurations with a non-negative weight function w . The generating function for S with respect to w is Φ S ( x ) = X σ S x w ( σ ) . Remark 3. By collecting terms with the same power of x , we obtain Φ S ( x ) = X k 0 a k x k , where a k is the number of configurations of weight k .
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