Combinatorics Through Guided Discovery, 2004a

More documents

Recommendations

Info

189 286. Before you try to show that σ actually is a permutation of the colorings, it would be useful to verify the second part of the definition of a group action, namely that σ ◦ ϕ = σ ◦ ϕ. 289. If z ∈ Gx and z ∈ Gy , how can you use elements of G to explain the relationship between x and y? Additional Hint: Suppose σ is a fixed member of G. As τ ranges over G, which elements of G occur as τσ? 295. How does the size of a multiorbit compare to the size of G? 301. We are asking for the number of orbits of some group on lists of four Rs, six Bs, and seven Gs. 305. There are five kinds of elements in the rotation group of the cube. For example, there are six rotations by 90 degrees or 270 degrees around an axis connecting the centers of two opposite faces and there are 8 rotations (of 120 degrees and 240 degrees, respectively) around an axis connecting two diagonally opposite vertices. 306. Is it possible for a nontrivial rotation to fix any coloring? 309. There are 48 elements in the group of automorphisms of the graph. Additional Hint: For this problem, it may be easier to ask which group elements fix a coloring rather than which colorings are fixed by a group element. 326. The group of automorphisms of the graph has 48 elements and contains D 6 as a subgraph. Additional Hint: The permutations with four one-cycles and the two-cycle (1 4), (2 5), or(3 6) are in the group of automorphisms. Once you know the cycle structure of D 6 and (1 4)D 6 = {(1 4)σ|σ ∈ D 6 }, you know the cycle structure of every element of the group. 327. What does the symmetric group on five vertices have to do with this problem? 329.c. In the relation of a function, how many pairs (x, f (x)) have the same x-value? 332. For the second question, how many arrows have to leave the empty set? How many arrows have to leave a set of size one? 339. What is the domain of g ◦ f ? 345. If we have scoops of vanilla, chocolate, and strawberry sitting in a circle in a dish, can we distinguish between VCS and VSC?
190 D. Hints to Selected Problems 349. To show a relation is an equivalence relation, you need to show it satisfies the definition of an equivalence relation. 353. To get you started, in Problem 38 the equivalence classes correspond to seating arrangements. 361. You’ve probably guessed that the sum is n 2 . To prove this by contradiction, you have to assume it is false, that is, that there is an n such that 1+3+ 5+···+2n − 1 n 2 . Then the method of Problem 360 says there must be a smallest such n and suggests we call it k. Why do you know that 1+3+5+···+2k − 3=(k − 1)2? What happens if you add 2n − 1 to both sides of the equation? 365. You’ve probably already seen that, with small values of n, sometimes n 2 and sometimes 2 n is bigger. But if you keep experimenting one of the functions seems to get bigger and stay bigger than the other. The number n = b where this change occurs is a good choice for a base case. So as not to spoil the problem for you, we won’t say here what this value of b is. However you shouldn’t be surprised later in the proof if you need to use the assumption that n/gtb. Additional Hint: You may have reached the point of assuming that 2 k−1 > (k − 1) 2 and found yourself wondering how to prove that 2 k > k 2 . A natural thing to try is multiplying both sides of 2 k−1 > (k − 1) 2 by 2. This ends up giving you 2 k > 2k 2 − 4k +2. Based on previous experience it is natural for you to expect to see how to turn this new right hand side into k 2 but not see how to do it. Here is the hint. You only need to show that the right hand side is greater than or equal to k 2 . For this purpose you need to show that one of the two k 2 sin2k 2 somehow balances out the −4k. See if you can figure out how the fact that you are only considering ks with k > b can help you out. 366. When you suspect an argument is not valid, it may be helpful to explicitly try several values of n to see if it makes sense for them. Often small values of n are adequate to find the flaw. If you find one flaw, it invalidates everything that comes afterwards (unless, of course, you can fix the flaw). 370. You might start out by ignoring the word unique and give a proof of the simpler theorem that results. Then look at your proof to see how you can include uniqueness in it. 377. An earlier problem may help you put your answer into a simpler form. 378. What is the power series representation of e x2 ? 381. There is only one element that you may choose. In the first case you either choose it or you don’t. 387.b. At some point, you may find the binomial theorem to be useful.
Page 2:
Combinatorics Through Guided Discov
Page 5 and 6:
Edition: First PreTeXt Edition Webs
Page 7 and 8:
vi a great deal of time to helping
Page 9 and 10:
viii because they use an important
Page 11 and 12:
x you happen to get a hard problem
Page 13 and 14:
xii Discovery had been placed on th
Page 15 and 16:
xiv Contents 2.3.1 Undirected graph
Page 17 and 18:
xvi Contents C Exponential Generati
Page 19 and 20:
2 1. What is Combinatorics? learned
Page 21 and 22:
4 1. What is Combinatorics? For exa
Page 23 and 24:
6 1. What is Combinatorics? and the
Page 25 and 26:
8 1. What is Combinatorics? + Probl
Page 27 and 28:
10 1. What is Combinatorics? If we
Page 29 and 30:
12 1. What is Combinatorics? are go
Page 31 and 32:
14 1. What is Combinatorics? k =0 1
Page 33 and 34:
16 1. What is Combinatorics? • Pr
Page 35 and 36:
18 1. What is Combinatorics? • Pr
Page 37 and 38:
20 1. What is Combinatorics? a Prov
Page 39 and 40:
22 1. What is Combinatorics? (c) Fi
Page 41 and 42:
24 1. What is Combinatorics? (vii)
Page 43 and 44:
26 1. What is Combinatorics? · Pro
Page 45 and 46:
28 1. What is Combinatorics? 1.4 Su
Page 47 and 48:
30 1. What is Combinatorics?
Page 49 and 50:
32 2. Applications of Induction and
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
52 3. Distribution Problems problem
Page 71 and 72:
54 3. Distribution Problems bĳecti
Page 73 and 74:
56 3. Distribution Problems 3.1.3 M
Page 75 and 76:
58 3. Distribution Problems broken
Page 77 and 78:
60 3. Distribution Problems 3.2.2 S
Page 79 and 80:
62 3. Distribution Problems coeffic
Page 81 and 82:
64 3. Distribution Problems Problem
Page 83 and 84:
66 3. Distribution Problems Problem
Page 85 and 86:
68 3. Distribution Problems number
Page 87 and 88:
70 3. Distribution Problems ⇒ Pro
Page 89 and 90:
72 3. Distribution Problems ⇒ 12.
Page 91 and 92:
74 4. Generating Functions • Prob
Page 93 and 94:
Page 95 and 96:
78 4. Generating Functions x k in t
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
84 4. Generating Functions the gene
Page 103 and 104:
86 4. Generating Functions ◦ Prob
Page 105 and 106:
88 4. Generating Functions Writing
Page 107 and 108:
90 4. Generating Functions 4.4 Supp
Page 109 and 110:
92 4. Generating Functions
Page 111 and 112:
94 5. The Principle of Inclusion an
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
100 5. The Principle of Inclusion a
Page 119 and 120:
102 5. The Principle of Inclusion a
Page 121 and 122:
104 6. Groups acting on sets on. We
Page 123 and 124:
106 6. Groups acting on sets still
Page 125 and 126:
108 6. Groups acting on sets Let us
Page 127 and 128:
110 6. Groups acting on sets Proble
Page 129 and 130:
112 6. Groups acting on sets 6.1.7
Page 131 and 132:
114 6. Groups acting on sets ⇒ ·
Page 133 and 134:
116 6. Groups acting on sets • Pr
Page 135 and 136:
118 6. Groups acting on sets 3, and
Page 137 and 138:
Page 139 and 140:
122 6. Groups acting on sets Taking
Page 141 and 142:
Page 143 and 144:
126 6. Groups acting on sets of dif
Page 145 and 146:
Page 147 and 148:
130 6. Groups acting on sets ⇒ Pr
Page 149 and 150:
132 6. Groups acting on sets (c) Wr
Page 151 and 152:
134 A. Relations Problem 329. Here
Page 153 and 154:
136 A. Relations Problem 333. Draw
Page 155 and 156: 138 A. Relations Problem 340. Expla
Page 157 and 158: 140 A. Relations • If R is a rela
Page 159 and 160: 142 A. Relations Problem 355. Suppo
Page 161 and 162: 144 A. Relations
Page 163 and 164: 146 B. Mathematical Induction (e) P
Page 165 and 166: 148 B. Mathematical Induction shows
Page 167 and 168: 150 B. Mathematical Induction cent
Page 169 and 170: 152 C. Exponential Generating Funct
Page 187 and 188: 170 D. Hints to Selected Problems 1
Page 195 and 196: 178 D. Hints to Selected Problems t
Page 197 and 198: 180 D. Hints to Selected Problems t
Page 201 and 202: 184 D. Hints to Selected Problems A
Page 205: 188 D. Hints to Selected Problems 2
Page 209 and 210: 192 D. Hints to Selected Problems
Page 211 and 212: 194 E. GNU Free Documentation Licen
Page 217 and 218: 200 Index equivalent, 112 cycle ind
Page 219 and 220: 202 Index pair structure, 159 pair,
show all

Combinatorics Through Guided Discovery, 2004a

Create successful ePaper yourself

Delete template?

Save as template?