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548 Week9

Course: MATH 548, Spring 2012
School: UNC
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For W9 (7.48) each of Parts (a) and (g), do only Steps (i) and (ii) below: Find the generating function H(x) for the sequence determined by the recurrence and the initial conditions, as on p. 235. (We will not discuss the partial fractions technique, which could then be used to find an explicit solution to the recurrence.) Here is an added part to this exercise: (g) hn = 7hn1 10hn2 for n 2 with h0 = 4 and h1 =...

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For W9 (7.48) each of Parts (a) and (g), do only Steps (i) and (ii) below: Find the generating function H(x) for the sequence determined by the recurrence and the initial conditions, as on p. 235. (We will not discuss the partial fractions technique, which could then be used to find an explicit solution to the recurrence.) Here is an added part to this exercise: (g) hn = 7hn1 10hn2 for n 2 with h0 = 4 and h1 = 7. (i) First find a rational function expression for H(x) whose numerator is expressed in terms of h0, h1, , hk1. (ii) Then use the specified initial conditions to rewrite the numerator as a specific polynomial in x. (Consider photocopying your work for use with Step (iii) of this exercise.) (8.81) Suppose we have two identical 3, three identical 5, and two identical 7 stamps available. Use a generating function to find all of the postage amounts which we can form using some of these stamps, and the number of different ways each possible postage amount can be formed. (8.82) Suppose we want to form a basket of fruit subject to the following requirements: The number of Apples must be even, there between are 3 and 17 Bananas inclusive, and the number of Cantaloupes is a multiple of 5 and nonzero. Further suppose that Apples cost 4, Bananas cost 5, and Cantaloupes cost 9. Find an identity for the generating function that describes how many baskets can be formed that have a total cost of n cents for n 0. (8.83) Let k 1. Find an identity for the generating function for the counts pk(n), n 0, of partitions of n into exactly k parts. (Hint: The diagrams for the partitions counted by pk(n) have at least one column of length k. So one approach would be to indicate how lecture's proof of the identity for the generating function for pk(n) can be slightly modified to produce a slightly different identity.) (8.25) Remark: My proof of the identity presented in lecture for the generating function for pk(n) tracked the numbers of columns of various lengths in the Ferrers diagrams of the partitions. Hint: Imitate that proof, but now track the numbers of rows of various lengths in the Ferrers diagrams of the partitions as I did when proving the generating function identity for partitions into odd parts.
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UNC - MATH - 548
W 10aBijective Proof Requirements: For n 0, let A n and B n be sets ofcombinatorial objects. Set an := |An| and bn := |Bn|. If you are asked toprovide a bijective proof that an = bn for n 0, you are to do the followingfor general n 0: Define a functio
UNC - MATH - 548
UNC - MATH - 548
UNC - MATH - 548
UNC - MATH - 548
UNC - MATH - 548
UNC - MATH - 548
UNC - MATH - 548
W 10b/11a( V 3 41/2) Consider the bipartite graph G = (X,Y) in whichX = cfw_1,2,3,8,9, Y = cfw_a,b,c,h,i, and has many edges. In particular,the edge set contains both the matchingM = cfw_(1,a), (3,c), (5,i), (6,f), (7,d), (8,e), (9,h) and the matching
UNC - MATH - 548
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