By David Guichard

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**Extra info for An Introduction to Combinatorics and Graph Theory**

**Example text**

Show that the Maclaurin series for (x + 1)r is ∞ i=0 r i xi . 3. 4, show that all coefficients beginning with x16 are 180. 4. Use a generating function to find the number of solutions to x1 + x2 + x3 + x4 = 14, where 0 ≤ x1 ≤ 3, 2 ≤ x2 ≤ 5, 0 ≤ x3 ≤ 5, 4 ≤ x4 ≤ 6. 5. Find the generating function for the number of solutions to x1 + x2 + x3 + x4 = k, where 0 ≤ x1 ≤ ∞, 3 ≤ x2 ≤ ∞, 2 ≤ x3 ≤ 5, 1 ≤ x4 ≤ 5. 6. Find a generating function for the number of non-negative integer solutions to 3x + 2y + 7z = n.

1. Prove that r k = r−1 k−1 + r−1 k . 2. Show that the Maclaurin series for (x + 1)r is ∞ i=0 r i xi . 3. 4, show that all coefficients beginning with x16 are 180. 4. Use a generating function to find the number of solutions to x1 + x2 + x3 + x4 = 14, where 0 ≤ x1 ≤ 3, 2 ≤ x2 ≤ 5, 0 ≤ x3 ≤ 5, 4 ≤ x4 ≤ 6. 5. Find the generating function for the number of solutions to x1 + x2 + x3 + x4 = k, where 0 ≤ x1 ≤ ∞, 3 ≤ x2 ≤ ∞, 2 ≤ x3 ≤ 5, 1 ≤ x4 ≤ 5. 6. Find a generating function for the number of non-negative integer solutions to 3x + 2y + 7z = n.

N! n=0 n=0 n k=0 n Bn−k k xn , n! 4. Now it is possible to write this as a product of two infinite series: ∞ Bn · f (x) = n=0 xn n! ∞ an xn = f (x)g(x). n=0 Find an expression for an that makes this true, which will tell you what g(x) is, then solve the differential equation for f (x), the exponential generating function for the Bell numbers. 4, the first few Bell numbers are 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437. You can use Sage to check your answer. 1 A partition of a positive integer n is a multiset of positive integers that sum to n.