By David Guichard
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Extra info for An Introduction to Combinatorics and Graph Theory
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6. 1 How many submultisets of the multiset {2 · a, 4 · b, 3 · c} have size 7? We recast the problem: this is the number of solutions to x1 + x2 + x3 = 7 with 0 ≤ x1 ≤ 2, 0 ≤ x2 ≤ 4, 0 ≤ x3 ≤ 3. We know that the number of solutions in nonnegative integers is 7+3−1 = 92 , so this is an overcount, since we count solutions that 3−1 do not meet the upper bound restrictions. For example, this includes some solutions with x1 ≥ 3; how many of these are there? This is a problem we can solve: it is the number of solutions to x1 + x2 + x3 = 7 with 3 ≤ x1 , 0 ≤ x2 , 0 ≤ x3 .
4, or your favorite computer algebra system. 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.
Using the usual convention that an empty sum is 0, we say that p0 = 1. 2 The partitions of 5 are 5 4+1 3+3 3+1+1 2+2+1 2+1+1+1 1 + 1 + 1 + 1 + 1. Thus p5 = 7. There is no simple formula for pn , but it is not hard to find a generating function for them. As with some previous examples, we seek a product of factors so that when the factors are multiplied out, the coefficient of xn is pn . We would like each xn term to represent a single partition, before like terms are collected. A partition is uniquely described by the number of 1s, number of 2s, and so on, that is, by the repetition numbers of the multiset.