By David Guichard

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**Extra resources 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.

So when i is large, √ 1 1+ 5 i Fi = round √ , 2 5 √ 5)/2| < 1, the limit of that is, the first term rounded to the nearest integer. As it turns out, this is true starting with i = 0. You can see how to do the entire solution in Sage. 4. 1. Find the generating function for the solutions to hn = 4hn−1 − 3hn−2 , h0 = 2, h1 = 5, and use it to find a formula for hn . 2. Find the generating function for the solutions to hn = 3hn−1 + 4hn−2 , h0 = h1 = 1, and use it to find a formula for hn . 3. Find the generating function for the solutions to hn = 2hn−1 + 3n , h0 = 0, and use it to find a formula for hn .

N (−1)n−j j=1 n−1 k j−1 j−1 k (−1)n−j−1 n−1 j n−1 =k j=1 n (−1)n−j (n − 1) + j=1 j k n−1 j n−1 − (n − 1) (−1)n−j−1 j=1 j k n−1 j j k = kδn−1,k − (n − 1)δn−1,k . Now if k = n − 1, this is (n − 1)δn−1,n−1 − (n − 1)δn−1,n−1 = 0, while if k < n − 1 it is kδn−1,k − (n − 1)δn−1,k = k · 0 − (n − 1) · 0 = 0. If we interpret the triangles containing the s(n, k) and S(n, k) as matrices, either m × m, by taking the first m rows and columns, or even the infinite matrices containing the entire triangles, the sums of the theorem correspond to computing the matrix product in both orders.

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

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