Bergeron F., Labelle G., Leroux P. — Combinatorial Species and Tree-like Structures :: Электронная библиотека попечительского совета мехмата МГУ

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Bergeron F., Labelle G., Leroux P. — Combinatorial Species and Tree-like Structures

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Название: Combinatorial Species and Tree-like Structures

Авторы: Bergeron F., Labelle G., Leroux P.

Аннотация:

The combinatorial theory of species, introduced by Joyal in 1980, provides a unified understanding of the use of generating functions for both labelled and unlabelled structures and as a tool for the specification and analysis of these structures. Of particular importance is their capacity to transform recursive definitions of tree-like structures into functional or differential equations, and vice versa. The goal of this book is to present the basic elements of the theory and to give a unified account of its developments and applications. It offers a modern introduction to the use of various generating functions, with applications to graphical enumeration, Polya theory and analysis of data structures in computer science, and to other areas such as special functions, functional equations, asymptotic analysis and differential equations. This book will be a valuable reference to graduate students and researchers in combinatorics, analysis, and theoretical computer science.

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Рубрика: Математика/Алгебра/Комбинаторика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 457

Добавлена в каталог: 10.03.2005

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Предметный указатель

$Bal^{[k]}$ , k level ballots      36
$Cov^{[m]}$ , m parts coverings      78
$C^{[m]}$ , m-complexes      78
$End^{\wp}$ , partial endofunctions      172
$End^{\wp}_{R}$ , R-enriched partial endofunctions      172
$End_{R}$ , R-enriched endofunctions      172
$E^{\pm}$ , oriented sets      144
$Oct_{alt}$ , alternating octopuses      113
$Oct_{reg}$ , regular octopuses      56
$Par^{[k]}$ , partitions with k parts      31
$Par_{P}$ , partial set partitions      49
$Rel^{[m]}$ , m-ary relations      74
$\Gamma(X,Y)$ , graphs on vertices and edges      111
$\Lambda^{\alpha}$ , connected components weighting species      98
$\mathbb{B}$ -species      341 345
$\mathbb{L}$ -species      341 344
$\mathbb{L}$ -species, multisort      353
$\mathbb{L}$ -species, weighted      344
$\mathcal{C}$ , cyclic permutations      7
$\mathcal{C}_{alt}$ , alternating cyclic permutations      108
$\mathcal{D}$ , directed graphs      7
$\mathcal{F}$ , forests      52
$\mathcal{G}$ , simple graphs      4
$\mathcal{G}^{c}$ , connected simple graphs      27
$\mathcal{G}^{d}$ , disconnected simple graphs      27
$\mathcal{H}$ , hedges of rooted trees      10
$\mathcal{H}$ , homeomorphically irreducible rooted trees      283
$\mathcal{K}_{n}$ , complete graphs      297
$\mathcal{P}$ , commutative parenthesizations      10
$\mathcal{P}$ , plane rooted trees      168
$\mathcal{S}$ , permutations      7
$\mathcal{S}^{<k>}$ , permutations with all cycles of k      40
$\mathcal{S}^{mix}$ , mixed permutations      114
$\mathcal{S}^{tric}$ , tricolored permutations      107
$\mathcal{S}^{[k]}$ , permutations with k cycles      31
$\mathcal{V}$ , vertebrates      61
$\mathfrak{h}$ , homeomorphically irreducible trees      283
$\mathfrak{p}$ , plane trees      178
$\Phi$ , functions      112
$\varepsilon$ , elements      8
$\wp$ , subsets      7
$\wp^{[k]}$ , k-element subsets      34
(2,3)-rooted trees      198 211 230
(2,3)-trees      230 244
(a,b)-trees      243
0, empty species      8
1, empty set species      8
2-Connected graphs      299—303
2-edge connected graphs      301 307
A-weighted sets      81
A-weighted species      82
Abel identities      189
Admissible formal power series      257
Alcohol molecules      287 288
Algebraic identities (extension principle)      182
Algebraic singularity      253
Alkane molecules      287 288
Alternating cycles      105 108
Alternating group      144 321
Alternating octopuses      113
Alternating permutations      344 345 351
ascent      355
Assembly of structures      42
Asymmetric assemblies      328
Asymmetric circular assemblies      328
Asymmetric colored cycles      93
Asymmetric colored permutations      93
Asymmetric permuted assemblies      328
Asymmetric plane rooted trees      336
Asymmetric plane trees      336
Asymmetric rooted trees      322 329—330 338
Asymmetric species      322
Asymmetric structures      322 324
Asymmetric trees      321 329—330
Asymmetry, generating series      324
Asymmetry, index series      324 327
Asymptotic      247
Asymptotic expansion      255—256
Asymptotic scale      255
Asymptotically equivalent      248
Atomic decomposition      147 150
Atomic k-sort species      158
Atomic species      146 154 329 430—431
Automorphism      3 6
AVL trees      238 245
B-trees      243
Ballots      10 36 44 57 69 124 251 254 309 412 413 415 417
bc-tree of a graph      300—301
Bell numbers      63 116
Bender's theorem      261—262 274
Bernoulli numbers      268
Bernoulli polynomials      268
Bicolored plane trees      215—217
Bicolored rooted plane trees      215
Bicolored rooted trees      206
Bicolored rooted, R, S-enriched      206
Bicolored trees      206
Bicolored trees, R, S-enriched      206 294
Bicolored vertebrates      215
Bij, bijections      112
Billiard stroke      237
Binary (plane) rooted trees      8 10 162 195 235 261 373
Binary (plane) rooted trees, complete      235
Binary (plane) rooted trees, complete, increasing      345 358
Binary (plane) rooted trees, height of      235
Binary (plane) rooted trees, increasing      344 345 359 373 380
Binary (plane) rooted trees, leftist      246
Binary (plane) rooted trees, pseudo leftist      246
Binary (plane) rooted trees, Strahler order of      235
Binary (plane) rooted trees, unary      254
binary search trees      341
Binomial type sequences      169 181 187 188
Block of a graph      300
Block-cut-point tree      300
Bridge of a graph      307
Buds      361 365
Burnside's Lemma      396
c-chromatic trees      294
c-chromatic trees, rooted      294
Cancellation law for addition      134
Canonical decomposition      30 39 109 123
Cartesian product of, (weighted) L-species      348
Cartesian product of, species      64 410 413 414 416 418
Cartesian product of, virtual species      121
Cartesian product of, weighted species      84
Catalan numbers      163 168 252 256 291
Cauchy — Frobenius theorem (Burnside's lemma)      396
Cayley's formula      167 180
Cayley's identity      290
Center of a tree      280
Center of a tree of mass      290
Central trinomial coefficients      183
Cha, chains      11
Chains      11 56 160 297 409 412 414 415 417
Characterization of atomic species      154
Circular words      92
Colorations      404
Colored cycles      93
Colored derangements      96
Colored permutations      93
Colorings      399
Colorings, F-injective      325
Colorings, G-reduced      318 399
Combinatorial differential equations      358 363
Combinatorial eclosions      212 365
Combinatorial equality      21
Combinatorial exponential      131
Combinatorial logarithm      130 149
Combinatorial separation of variables      377
Commutative parenthesizations      10 198 222 241 379 410 412 414

Commuting group actions      407
Compatible permutations      218
Complementary subspecies      134
Complete binary rooted trees      235
Complete graphs      297
Complete increasing binary rooted trees      345 358 373 380
Composite (functorial)      73
Composite (partitional) of L-species      348 353
Composite (partitional) of species      41 104
Composition of cycle index series      43 73 84 107
Concentrated (species) on a cardinality      30 140
Conjugate subgroups      395
Conjugate words      92
Connected compatible permutations      219
Connected components weighting species      99 340
Connected linear orders, lists      131 348
Connected R-enriched endofunctions      202
Connected simple graphs      7 37 46 64 297 298
Connected structures      46 55 63 96 131 159 308 410 412 414 416 418
Contact order      22 23 194
Convolution      348—349 357
Counter for arcs      411
Counter for connected components      97 411 413—418
Counter for cycles      83 86 91 184 411
Counter for edges      112 297 308 405 411
Counter for elements      100 411
Counter for fixed points      89 411
Counter for leaves      80
Counter for parts      86 411
Counter for transpositions      89 411
Cov, coverings      78
Coverings of sets      78
Cubic anharmonic oscillator      389
Cut-point of a graph      299
Cycle counter      83 86 91 184 411
Cycle index, polynomial      397
Cycle index, series      17 84 106 122 207—209 310
Cycle type of permutations      16
Cyclic group      406
Cyclic permutations, oriented cycles      2 7 45 48 58 86 92 109 133 144 155 190 220 293 322 346 409 411 413 415 417
Cyclic permutations, oriented cycles, alternating      105 108
Cyclic permutations, oriented cycles, colored      93
Cyclic permutations, oriented cycles, colored, asymmetric      93
Cyclic rooted trees      see “Mobiles”
Cyclotomic identities      98
Degenerate vertebrates      62
Der, derangements      10
Derangements      10 31 32 33 39 220 250 409 412 413 415 417
Derangements, colored      96
Derivation of (weighted) L-species      348
Derivation of species      47
Derivation of virtual species      125
Derivation of weighted species      84
Derivation, generalized      119
Derivation, partial      105 354
Descent of a permutation (list)      346
Differential equations, autonomous      360
Differential equations, combinatorial      358 363
Differential equations, higher order      371
Differential equations, linear      377
Differential equations, non autonomous      371 377
Differential equations, system of      358 371
Differential operators      365 372
Differential operators, powers of      368 386
Dihedral group      144 321 406
Directed, graphs      7 74 76 77 409 411 415
Directed, trees      292
Disconnected simple graphs      27
Discrete graphs      297
Dissections      104
Dissymmetry theorem, dissymmetry formula, for asymmetric trees      330
Dissymmetry theorem, dissymmetry formula, for c-chromatic trees      294
Dissymmetry theorem, dissymmetry formula, for graphs      301
Dissymmetry theorem, dissymmetry formula, for plane trees      291
Dissymmetry theorem, dissymmetry formula, for R,S-enriched trees      207 294
Dissymmetry theorem, dissymmetry formula, for R-enriched trees      282 291
Dissymmetry theorem, dissymmetry formula, for R-enriched trees with leaves      288
Dissymmetry theorem, dissymmetry formula, for trees      277 280 296
Dobinski's formula      116
Dominant singularities      249
Double cosets      153 328
Duffing equation      389
E, sets      8
Eccentricity of a vertex      280
Eclosion      212
Eclosion points      366
Edge counter      112 297 308 405
Edges      72
Electric circuit with a quadratic resistance      388
Elementary symmetric functions      333
Elements      8 409 412 413 415 417
Empty set species      8 346 409
Empty species      8 346 409
End, endofunctions      7
Endofunctions      7 41 62 63 86 209 328 412 414
Endofunctions, idempotent      221
Endofunctions, Jacobi      172 203 245
Endofunctions, partial      172
Endofunctions, R-enriched      172 184 200 218 220 410
Endofunctions, R-enriched, connected      202
Endofunctions, R-enriched, partial      172 200
Equipotence of species      20
Euler constant      267
Euler gamma function      266
Euler numbers      352 383
Euler — Maclaurin summation formula      268
Eulerian numbers      354
Eulerian polynomials      354
Even alternating permutations (lists)      351 359
Even complete increasing binary rooted trees      359
Even graphs      351
Even part of species      38 410 412 414 416—417
Even permutations      144 337
Even sets      29 348 409 411 413 415 416
Excedance      355
Exponential formula (weighted)      96
Exponential generating series      13
Extension principle of algebraic identities      182
F-injective assemblies      325
F-injective colorings      325
F-MacLaurin expansion      118
F-structures      5 102 345
F-Taylor expansion      118
Falling factorial      170
Fatou's formula      233
Fibers      113 165 363
Fibonacci numbers      249
Flat part      322—323
Flat species      322
Fliess series      389
Forests      52 58 170 260 274 293
Frobenius formula      354
Functional digraph      7
Functions with R-enriched fibers      113 170
Functorial composition of species      70 71 410 413—414 416 418
Functorial composition of virtual species      125
Functorial composition of weighted multisort species      105
Functorial composition of weighted species      84
Functoriality      5 11
Fundamental transformation      22 174
G-eclosions      212
G-rooted trees      195 197 213 229 262 275
G-rooted trees, unlabeled      263 276
Gamma function      266
Genealogical bees      188
Generalized differentiation      119
Generalized pointing      117
Generalized separation of variables      387
Generating series      13 83 106 122 346
Generating series, asymmetry type      324

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