proofs

Recommendations

Info

Identities versus bijections 211 As experienced formal series manipulators we notice that the introduction of the new variable y yields ∏ (1 + yx k ) = ∑ p d,m (n)x n y m , k≥1 n,m≥0 where p d,m (n) counts the partitions of n into precisely m distinct summands. With y = −1 this yields ∏ − x k≥1(1 k ) = ∑ (E d (n) − O d (n))x n , (10) n≥0 where E d (n) is the number of partitions of n into an even number of distinct parts, and O d (n) is the number of partitions into an odd number. And here is the punchline. Comparing (10) to Euler’s expansion in (8) we infer the beautiful result ⎧ 1 for n = ⎪⎨ 3j2 ±j 2 when j ≥ 0 is even, E d (n) − O d (n) = −1 for n = 3j2 ±j 2 when j ≥ 1 is odd, ⎪⎩ 0 otherwise. An example for n = 10: 10 = 9 + 1 10 = 8 + 2 10 = 7 + 3 10 = 6 + 4 10 = 4 + 3 + 2 + 1 and 10 = 10 10 = 7 + 2 + 1 10 = 6 + 3 + 1 10 = 5 + 4 + 1 10 = 5 + 3 + 2, so E d (10) = O d (10) = 5. This is, of course, just the beginning of a longer and still ongoing story. The theory of infinite products is replete with unexpected indentities, and with their bijective counterparts. The most famous examples are the so-called Rogers–Ramanujan identities, named after Leonard Rogers and Srinivasa Ramanujan, in which the number 5 plays a mysterious role: ∏ k≥1 1 (1 − x 5k−4 )(1 − x 5k−1 ) = ∑ n≥0 x n2 (1 − x)(1 − x 2 ) · · · (1 − x n ) , ∏ k≥1 1 (1 − x 5k−3 )(1 − x 5k−2 ) = ∑ n≥0 x n2 +n (1 − x)(1 − x 2 ) · · · (1 − x n ) . The reader is invited to translate them into the following partition identities first noted by Percy MacMahon: Srinivasa Ramanujan • Let f(n) be the number of partitions of n all of whose summands are of the form 5k + 1 or 5k + 4, and g(n) the number of partitions whose summands differ by at least 2. Then f(n) = g(n). • Let r(n) be the number of partitions of n all of whose summands are of the form 5k + 2 or 5k + 3, and s(n) the number of partitions whose parts differ by at least 2 and which do not contain 1. Then r(n) = s(n). All known formal series proofs of the Rogers–Ramanujan identities are quite involved, and for a long time bijection proofs of f(n) = g(n) and of r(n) = s(n) seemed elusive. Such proofs were eventually given 1981 by Adriano Garsia and Stephen Milne. Their bijections are, however, very complicated — Book proofs are not yet in sight.
212 Identities versus bijections References [1] G. E. ANDREWS: The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley, Reading MA 1976. [2] D. BRESSOUD & D. ZEILBERGER: Bijecting Euler’s partitions-recurrence, Amer. Math. Monthly 92 (1985), 54-55. [3] A. GARSIA & S. MILNE: A Rogers–Ramanujan bijection, J. Combinatorial Theory, Ser. A 31 (1981), 289-339. [4] S. RAMANUJAN: Proof of certain identities in combinatory analysis, Proc. Cambridge Phil. Soc. 19 (1919), 214-216. [5] L. J. ROGERS: Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318-343.
Page 2:
Martin Aigner Günter M. Ziegler Pr
Page 5 and 6:
Prof. Dr. Martin Aigner FB Mathemat
Page 7 and 8:
VI Preface to the Fourth Edition Wh
Page 9 and 10:
VIII Table of Contents 21. A theore
Page 12 and 13:
Six proofs of the infinity of prime
Page 14 and 15:
Six proofs of the infinity of prime
Page 16 and 17:
Bertrand’s postulate Chapter 2 We
Page 18 and 19:
Bertrand’s postulate 9 times. Her
Page 20 and 21:
Bertrand’s postulate 11 by compar
Page 22 and 23:
Binomial coefficients are (almost)
Page 24 and 25:
Binomial coefficients are (almost)
Page 26 and 27:
Representing numbers as sums of two
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
The law of quadratic reciprocity Ch
Page 34 and 35:
The law of quadratic reciprocity 25
Page 36 and 37:
Page 38 and 39:
Page 40 and 41:
Every finite division ring is a fie
Page 42 and 43:
Every finite division ring is a fie
Page 44 and 45:
Some irrational numbers Chapter 7
Page 46 and 47:
Some irrational numbers 37 and this
Page 48 and 49:
Some irrational numbers 39 F(x) may
Page 50:
Some irrational numbers 41 Now assu
Page 53 and 54:
44 Three times π 2 /6 v 1 1 2 y v
Page 55 and 56:
46 Three times π 2 /6 For m = 1, 2
Page 57 and 58:
48 Three times π 2 /6 1 f(t) = 1 t
Page 59 and 60:
50 Three times π 2 /6 (3) It has b
Page 62 and 63:
Hilbert’s third problem: decompos
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70:
Page 73 and 74:
64 Lines in the plane, and decompos
Page 75 and 76:
66 Lines in the plane, and decompos
Page 78 and 79:
The slope problem Chapter 11 Try fo
Page 80 and 81:
The slope problem 71 • Every move
Page 82:
The slope problem 73 For the second
Page 85 and 86:
76 Three applications of Euler’s
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
82 Cauchy’s rigidity theorem Q: Q
Page 93 and 94:
84 Cauchy’s rigidity theorem A be
Page 95 and 96:
86 Touching simplices l l l ′ Pr
Page 97 and 98:
88 Touching simplices For our examp
Page 99 and 100:
90 Every large point set has an obt
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
96 Borsuk’s conjecture Theorem. L
Page 107 and 108:
98 Borsuk’s conjecture On the oth
Page 109 and 110:
100 Borsuk’s conjecture To obtain
Page 112 and 113:
Sets, functions, and the continuum
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
In praise of inequalities Chapter 1
Page 130 and 131:
In praise of inequalities 121 where
Page 132 and 133:
In praise of inequalities 123 Proo
Page 134:
In praise of inequalities 125 Seco
Page 137 and 138:
128 The fundamental theorem of alge
Page 140 and 141:
One square and an odd number of tri
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
A theorem of Pólya on polynomials
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
On a lemma of Littlewood and Offord
Page 156 and 157:
On a lemma of Littlewood and Offord
Page 158 and 159:
Cotangent and the Herglotz trick Ch
Page 160 and 161:
Cotangent and the Herglotz trick 15
Page 162 and 163:
Cotangent and the Herglotz trick 15
Page 164 and 165:
Buffon’s needle problem Chapter 2
Page 166 and 167:
Buffon’s needle problem 157 The c
Page 168:
Combinatorics 25 Pigeon-hole and do
Page 171 and 172: 162 Pigeon-hole and double counting
Page 182 and 183: Tiling rectangles Chapter 26 Some m
Page 184 and 185: Tiling rectangles 175 Third proof.
Page 186: Tiling rectangles 177 References [1
Page 189 and 190: 180 Three famous theorems on finite
Page 191 and 192: 182 Three famous theorems on finite
Page 194 and 195: Shuffling cards Chapter 28 How ofte
Page 196 and 197: Shuffling cards 187 Indeed, if A i
Page 198 and 199: Shuffling cards 189 Let T be the nu
Page 200 and 201: Shuffling cards 191 Theorem 1. Let
Page 202 and 203: Shuffling cards 193 Proof. (1) We
Page 204 and 205: Lattice paths and determinants Chap
Page 206 and 207: Lattice paths and determinants 197
Page 208 and 209: Lattice paths and determinants 199
Page 210 and 211: Cayley’s formula for the number o
Page 216 and 217: Identities versus bijections Chapte
Page 218 and 219: Identities versus bijections 209 Pr
Page 222 and 223: Completing Latin squares Chapter 32
Page 224 and 225: Completing Latin squares 215 Proof
Page 226 and 227: Completing Latin squares 217 be dis
Page 228: Graph Theory 33 The Dinitz problem
Page 231 and 232: 222 The Dinitz problem In 1976 Vizi
Page 233 and 234: 224 The Dinitz problem X A bipartit
Page 235 and 236: 226 The Dinitz problem G : L(G) : a
Page 237 and 238: 228 Five-coloring plane graphs From
Page 239 and 240: 230 Five-coloring plane graphs is s
Page 241 and 242: 232 How to guard a museum The follo
Page 243 and 244: 234 How to guard a museum “Museum
Page 245 and 246: 236 Turán’s graph theorem Let us
Page 247 and 248: 238 Turán’s graph theorem Thus b
Page 250 and 251: Communicating without errors Chapte
Page 252 and 253: Communicating without errors 243 cl
Page 254 and 255: Communicating without errors 245 Le
Page 256 and 257: Communicating without errors 247 On
Page 258 and 259: Communicating without errors 249 No
Page 260 and 261: The chromatic number of Kneser grap
Page 262 and 263: The chromatic number of Kneser grap
Page 264: The chromatic number of Kneser grap
Page 267 and 268: 258 Of friends and politicians w 1
Page 270 and 271:
Probability makes counting (sometim
Page 272 and 273:
Page 274 and 275:
Page 276 and 277:
Page 278 and 279:
Proofs from THE BOOK 269
Page 280 and 281:
Index acyclic directed graph, 196 a
Page 282 and 283:
Index 273 matrix-tree theorem, 203
show all

proofs

Create successful ePaper yourself

Delete template?

Save as template?