An Introduction to Combinatorics and Graph Theory by David Guichard PDF

By David Guichard

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N+i−1 n+i−1 = (−1)i = (−1)i . i n−1 Thus ∞ ∞ n+i−1 −n i n+i−1 i (x + 1) = (−1) x = (−x)i . n−1 n−1 i=0 i=0 Now replacing x by −x gives −n (1 − x) ∞ = i=0 −n So (1 − x) is the generating function for 1, ∞ · 2, . . , ∞ · n} of size i. n+i−1 i x. 1 Newton’s Binomial Theorem 53 In many cases it is possible to directly construct the generating function whose coefficients solve a counting problem. 3 Find the number of solutions to x1 + x2 + x3 + x4 = 17, where 0 ≤ x1 ≤ 2, 0 ≤ x2 ≤ 5, 0 ≤ x3 ≤ 5, 2 ≤ x4 ≤ 6.

2. 1. Find the coefficient of x9 /9! 1. You may use Sage or a similar program. 2. Find an exponential generating function for the number of permutations with repetition of length n of the set {a, b, c}, in which there are an odd number of a s, an even number of b s, and an even number of c s. 3. Find an exponential generating function for the number of permutations with repetition of length n of the set {a, b, c}, in which the number of a s is even and at least 2, the number of b s is even and at most 6, and the number of c s is at least 3.

The sequence for the number of pairs of rabbits now starts out h0 = 1, h1 = 1, h2 = 3, h3 = 5, h4 = 11. Set up and solve a recurrence relation for the number of pairs of rabbits. Show also that the sequence statisfies hn = 2hn−1 + (−1)n . 5 Catalan Numbers A rooted binary tree is a type of graph that is particularly of interest in some areas of computer science. 1. The root is the topmost vertex. The vertices below a vertex and connected to it by an edge are the children of the vertex. It is a binary tree because all vertices have 0, 1, or 2 children.

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