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Andrews G.E. — The Theory of Partitions
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Название: The Theory of PartitionsАвтор: Andrews G.E. Аннотация: This book develops the theory of partitions. Simply put, the partitions of a number are the ways of writing that number as sums of positive integers. For example, the five partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. Surprisingly, such a simple matter requires some deep mathematics for its study. This book considers the many theoretical aspects of this subject, which have in turn recently found applications to statistical mechanics, computer science and other branches of mathematics.With minimal prerequisites, this book is suitable for students as well as researchers in combinatorics, analysis, and number theory.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1976Количество страниц: 255Добавлена в каталог: 06.12.2005Операции: Положить на полку |
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70—71 (k, i)-negative oscillation 143—145 152—154 (k, i)-positive oscillation 143—145 152—154 Abel's lemma 209 Alder polynomials 118—119 Alder,H.L. 118 Algebraic geometry 199—200 Andrews, G.E. 14 15 31 32 52 85 86 118 119 137 138 158 177 178 200 210 211 228 243 244 Arkin, J. 85 86 Askey, R. 51 52 Atkin, A.O.L. 139 143 158 160 161 175 177 178 200 Auluck, F.C. 100 101 228 Ayoub, R. xiii 85 86 Bachman P. xiii Bailey, W.N. 20 30 31 32 Barton, D.E. 243 244 Basic hypergeometric series 17 202 227 Basic hypergeometric series, Heine's fundamental transformation 19 Basic hypergeometric series, well-poised 106 Bateman, P.T. 101 Bell numbers 214 2]6 Bell polynomials 204—206 210 216 Bell, E.T. 204 205 206 210 211 214 Bender, E. 3 15 200 227 228 Berndt, B. 71 85 86 Binary partitions see Partition Binary representation 116 Binomial series 54 177 see Bipartite number 207 BLOCKS 215—217 219—220 222—223 Bohr, N. 227 228 Bratley, P. 200 Brigham, N.A. 100 101 Brylawski, T. 227 228 Bumping 185 187 Burnell, D. 243 244 Canonical basis 213—214 Carlitz, L. 31 32 52 65 66 179 183 200 202 207 210 211 228 Cauchy's integral theorem 72 93 Cauchy, A. 17 21 30 32 51 52 Cayley, A. 65 66 85 86 209 Chaundy. T.W. 200 Cheema's theorem 203 Cheema, M.S. 119 203 210 211 243 244 Chowla, S. 160 Chrystal, G. xiii Chu — Vandermonde summation 51 61 64 227 see Churchhouse conjecture 160 165 177 Churchhouse, R.F. 160 165 177 178 243 244 Circle method 85 class 187—188 Combinatorics 199 212ff composition 54—58 63 65 Composition, algorithms 244 Composition, asymptotic formulas 65 68 Composition, graphical representation 55 Composition, multipartite number 54 57 179 202 Composition, vector 54 57 65 Comtet, L. xiii 14 15 210 211 Continued fraction see Ramanujan's coninued fraction Daum, J.A. 30 32 David, F.N. 243 244 Daykin, D.E. 138 Dedekind sum 72 Dedekind's see Derivative, composite function 204 Dickson, L.E. 227 228 243 244 Differential equations 132 Dillon, J.F. 65 66 Distributions 61—62 Dots 7 Doubilet, P. 212 227 228 Dragonette, L. 82 85 86 Durfee square 23 27—28 Durfee square, side of 28 Dyson, F.J. 142 158 161 175 178 Elliptic modular function see Modular function Elliptic theta functions see Theta funcions EQUIVALENCE see Partition Erdos, P. 56 65 66 101 Erlangen, University of 137 Euler pair 138 Euler's partition theorem 24 26 122—123 139 203 Euler's pentagonal number theorem see Pentagonal number theorem Euler, L. xiii 5 14 15 19 21 24 26 29 30 31 32 123 139 203 Eulerian series see Basic hypergeometric series F arey fractions 72 Faa di Bruno's formula 205 210 Factorizations, unordered 210 Fall 43 Farey dissection 72 94 Ferrers graph see Partition Fibonacci numbers 63 137 Fine, N.J. 26 29 31 32 210 211 Foata, D. 45 51 52 65 66 67 Ford circles 85 Formal power series 3 FORTRAN IV 230 Foulkes, H.O. 67 Fractional part 83 Franklin, F. 9 15 52 Fray, R.D. 67 Fruchl, R. 227 228 Fulton, J.D. 227 228 Fundamental partition ideal problem 123 Gamma function, Euler's integral 83 Gauss, C.F. 23 31 32 35 51 52 Gaussian multinomial coefficient 39 Gaussian polynomial 33 35—39 49 146 180 212 214 240 Gaussian sum 37 Generating function 221 see Generating function, algorithms from 233 Generating function, Bell polynomials 205 Generating function, determinants 179 Generating function, finite product 42 Generating function, infinite product 3 88 203—204 209 Generating function, multipartite partitions 203 Generating function, nonmodular 85 Generating function, plane partition 185 199 210 Generating function, recurrences 179—180 Generating function, two-variable 16 128 Geometric series, finite 4 Geometric series, infinite 4 Glaisher, J.W.L. 6 9 15 Goellnitz — Gordon identities 113—114 123 Goellnitz — Gordon identities, generalization 113—114 123 Goellnitz's theorem 117—118 Goldman, J. 227 229 Gollnitz, H. 105 113 114 118 119 121 123 Gordon's multipartite partition theorem 210—211 Gordon's theorem see Rogers — Ramanujan identities Gordon, B. 105 109 113 114 118 119 121 123 200 210 211 Greater index 42 Greatest integer function 83 157 Grosswald xiii 85 86 Group 198 Group theory 211 227 Group theory, statistical aspects 101 Group, cyclic 198 Gupta, H. 56 65 67 118 120 160 165 177 178 243 244 Gwyther, A.E. 243 244 Hagis, P. 81 82 85 86 Hahn, W. 31 32 Hankel's loop integral 75 79 Hardy, G.H. xiii 14 15 30 32 68 69 72 85 86 94 101 103 105 118 120 159 160 177 Haselgrove,C.B. 100 101 Hecke operator 160—161 167 Heine, E. 19 20 30 32 51 52 Hermite, C. 52 Hickerson, D.R. 15 Hindenburg algorithm 232 243 Hodge, W.V.D. 200 Hodges, J.H. 227 228 Homogeneous polynomial 204 Houten, L. 200 243 244 Hua, L.K. 82 86 Hurwitz zeta function 97 Hypercube, (k + 1) dimensional 198 Ideals 125 see Incidence algebra 217 Inclusion-exclusion 139 153 156 218 Inclusion-exclusion, generalized 218 Inclusion-exclusion, truncated 140 INDEX 236—237 Ingham, A.E. 100 102 Intersection 217 Invariant theory 199 inversion 40 -41 Iseki, S. 82 85 86 87 Jackson, F.H. 38 51 52 Jacobi's triple product identity seeTriple product identity Jacobi, C.G.J. 17 21 30 31 32 49 Join 122 217 Jordan, C. 227 228 Kalckar, F. 227 228 Kendall, M.G. 243 244 Kernel 219 Knopp, M.I. xiii 71 85 87 160 167 177 178 Knuth — Schensted correspondence 179 184—185 187 200 Knuth, D. 179 184 200 212 227 228 230 234 244 Knutson, D. 15 227 228 Kohlbecker, E.E. 100 102 Kolberg, O. 177 178 Kothari, D.S. 228 Kreweras, G. 65 67 199 201 Kronecker 217 Labeling 235 Landsberg, G. 227 228 Lascoux, A. 199 201 Lebesgue, V.A. 30 32 Legendre — Jacobi symbol 71 Lehmer, D.H. 86 101 102 Lehner, J. 56 65 66 85 86 101 120 177 178 LeVeque, W.J. 14 15 31 32 65 67 71 87 100 102 118 120 138 158 177 178 200 201 210 211 227 228 244 Lexicographic order 185 187 203 236 Lexicographic order, reverse 232 Linear transformation 225 Linearly independent vectors 213 Linked partition ideals see Partition Lipschitz summation formula 84 Littlewood, D.E. 227 228 Littlewood, J.E. 68 Livingood, J. 100 102 Logarithmic derivative 98 Long operator 64 Long, C. 65 67 MacDonald, I.G. 200 MacMahon, P.A. xiii 13 14 15 41 42 51 52 57 59 65 67 85 87 100 102 105 137 138 159 177 179 184 189 199 200 201 210 211 227 228 233 234 244 Matrices 184—185 187—188 Maximal element of 219 McKay, J.K.S. 200 Mediant 73 Meet 122 217 Meinardus, G. 88 89 99 100 101 102 Mellin transform 90 Menon, P.K. 31 32 Miller, J.C. p. 243 244 Miller, K.S. 132 138 Minimal element of 219 Mobius function 217?220 Mobius inversion 218 223—224 Mock-theta functions 29—30 82 Mock-theta functions, fifth-order 29—30 Modular equation 161 167 170 Modular form 70 160 177 Modular function 81—82 85—86 160 175 see Modular group 71 161 Modular transformation 71 82 Modulus 129—131 Moore, E. 15 Motzkin, T.S. 210 211 Multinomial coefficient 215 Multipartite number see also Partition Multipartite numbers 57 202—203 Multiset 39 126 Multiset, multiplicity in 39 Multiset, permutation 40—42 51 60—61 Murray — Miller algorithm 132 Murray, M. 132 138 Netto, E. 15 Newton's formula 168—169 Nijenhuis, A. 244 Order ideal see Partition Oscillation see (k i)-positive i) Ostmann, H.H. xiii 14 15 Parity of p(n) 233 Parkin, T.R. 244 Partial summation 98 Partially ordered set 220 235—237 Partially ordered set, infinite 23 7 Partially ordered set, locally finite 217 Partition 1 212 222 226 243 see Generatng Plane Sets Partition with congruence conditions 13 14 116—118 Partition with difference conditions 13 14 99 116 118 Partition with distinct parts 2 6 123 156 Partition with odd parts 2 5 13 14 123 237 Partition with parts +1 (mod 5) 109 237 Partition with parts - +2 (mod 5) 109 237 Partition, 51 65 200 227 244 Partition, algorithms 230 244 Partition, analysis 227 Partition, applications of 33 227 Partition, asymptotic formulas 56 68—70 88—89 93 97 Partition, binary 160—161 177 Partition, bipartite numbers 207 -209 243 Partition, Cayley's decomposition 81 Partition, computations for 220ff Partition, congruences 159—160 Partition, conjugate 7—9 28 Partition, F- 52 Partition, Ferrersgraph 6 13 40 142 214 Partition, flushed 31 Partition, from set H 3 16 100 Partition, functions 12 68 88 97 159—161 237 Partition, generating function 3—4 94 125 139 Partition, graphical representation 6—9 142 145 214 Partition, higher dimensional 179—180 189 197 199 230 234 244 Partition, higher dimensional, Ferrers graph 197—198 Partition, higher dimensional, graphical representation 197 199 Partition, history 244 Partition, ideal equivalence 122—123 128 Partition, ideals 121—125 129 Partition, ideals of order I 121 124—125 207 Partition, identities 17 18 118 121 125 202 207 233 Partition, in combinatorics 212ff Partition, into powers of m 175—176 Partition, k-dimensional 189 197—199 Partition, k-th excess 50 Partition, lattice structure 122 Partition, linked ideals 128 131 135 137 Partition, linking set 131 Partition, modular 13 14 Partition, multipartite number 57—58 202—204 206 210—21 Partition, multipartite number, distinct parts 203 Partition, multipartite number, odd parts 203 Partition, of infinity 137 Partition, order of ideal 124 Partition, P 235—237 Partition, partial ordering 122 Partition, parts 1
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