# Download e-book for kindle: An Introduction to Combinatorics and Graph Theory by David Guichard

By David Guichard

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A little thought leads to x −x e +e ∞ = i=0 ∞ xi (−x)i + = i! i! i=0 ∞ i=0 xi + (−x)i . i! Now xi + (−x)i is 2xi when i is even, and 0 when x is odd. Thus x −x e +e ∞ = i=0 2x2i , (2i)! 56 Chapter 3 Generating Functions so that ∞ i=0 x2i ex + e−x = . (2i)! 2 A similar manipulation shows that ∞ i=0 x2i+1 ex − e−x = . (2i + 1)! 2 Thus, the generating function we seek is ex − e−x ex + e−x x 1 1 e = (ex − e−x )(ex + e−x )ex = (e3x − e−x ). 5. 2. 1. Find the coefficient of x9 /9! 1. You may use Sage or a similar program.

N). This is a list of the numbers {1, . . , n} in some order, namely, this is a permutation according to our previous usage. We can continue to use the same word for both ideas, relying on context or an explicit statement to indicate which we mean. 6 and n n/2 (Sperner’s Theorem) The only anti-chains of largest size are n n/2 . Proof. First we show that no anti-chain is larger than these two. We attempt to partition n 2[n] into k = n/2 chains, that is, to find chains A1,0 ⊆ A1,1 ⊆ A1,2 ⊆ · · · ⊆ A1,m1 A2,0 ⊆ A2,1 ⊆ A2,2 ⊆ · · · ⊆ A2,m2 ..

I! i=0 ∞ i=0 xi + (−x)i . i! Now xi + (−x)i is 2xi when i is even, and 0 when x is odd. Thus x −x e +e ∞ = i=0 2x2i , (2i)! 56 Chapter 3 Generating Functions so that ∞ i=0 x2i ex + e−x = . (2i)! 2 A similar manipulation shows that ∞ i=0 x2i+1 ex − e−x = . (2i + 1)! 2 Thus, the generating function we seek is ex − e−x ex + e−x x 1 1 e = (ex − e−x )(ex + e−x )ex = (e3x − e−x ). 5. 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.

### An Introduction to Combinatorics and Graph Theory by David Guichard

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