Enumerative Combinatorics: Volume 2Cambridge University Press, 4 juin 2001 - 600 pages This second volume of a two-volume basic introduction to enumerative combinatorics covers the composition of generating functions, trees, algebraic generating functions, D-finite generating functions, noncommutative generating functions, and symmetric functions. The chapter on symmetric functions provides the only available treatment of this subject suitable for an introductory graduate course on combinatorics, and includes the important Robinson-Schensted-Knuth algorithm. Also covered are connections between symmetric functions and representation theory. An appendix by Sergey Fomin covers some deeper aspects of symmetric function theory, including jeu de taquin and the Littlewood-Richardson rule. As in Volume 1, the exercises play a vital role in developing the material. There are over 250 exercises, all with solutions or references to solutions, many of which concern previously unpublished results. Graduate students and research mathematicians who wish to apply combinatorics to their work will find this an authoritative reference. |
Table des matières
Trees and the Composition of Generating Functions | 1 |
52 Applications of the Exponential Formula | 10 |
53 Enumeration of Trees | 22 |
54 The Lagrange Inversion Formula | 36 |
55 Exponential Structures | 44 |
56 Oriented Trees and the MatrixTree Theorem | 54 |
Notes | 65 |
References | 69 |
79 A Scalar Product | 306 |
710 The Combinatorial Definition of Schur Functions | 308 |
711 The RSK Algorithm | 316 |
712 Some Consequences of the RSK Algorithm | 322 |
713 Symmetry of the RSK Algorithm | 324 |
714 The Dual RSK Algorithm | 331 |
715 The Classical Definition of Schur Functions | 334 |
716 The JacobiTrudi Identity | 342 |
Exercises | 72 |
Solutions to Exercises | 103 |
Algebraic DFinite and Noncommutative Generating Functions | 159 |
62 Examples of Algebraic Series | 168 |
63 Diagonals | 179 |
64 DFinite Generating Functions | 187 |
65 Noncommutative Generating Functions | 195 |
66 Algebraic Formal Series | 202 |
67 Noncommutative Diagonals | 209 |
Notes | 211 |
References | 214 |
Exercises | 217 |
Solutions to Exercises | 249 |
Symmetric Functions | 286 |
72 Partitions and Their Orderings | 287 |
73 Monomial Symmetric Functions | 289 |
74 Elementary Symmetric Functions | 290 |
75 Complete Homogeneous Symmetric Functions | 294 |
76 An Involution | 296 |
77 Power Sum Symmetric Functions | 297 |
78 Specializations | 301 |
717 The MurnaghanNakayama Rule | 345 |
718 The Characters of the Symmetric Group | 349 |
719 Quassymmetric Functions | 356 |
720 Plane Partitions and the RSK Algorithm | 365 |
721 Plane Partitions with Bounded Part Size | 371 |
722 Reverse Plane Partitions and the HillmanGrassl Correspondence | 378 |
723 Applications to Permutation Enumeration | 382 |
724 Enumeration under Group Action | 390 |
Notes | 396 |
References | 405 |
A1 Knuth Equivalence Jeu de Taquin and the LittlewoodRichardson Rule | 413 |
A12 Jeu de Taquin | 419 |
A13 The LittlewoodRichardson Rule | 429 |
Notes | 437 |
References | 438 |
A2 The Characters of GL n ℂ | 440 |
Exercises | 450 |
Solutions to Exercises | 490 |
561 | |
Additional Errata and Addenda | 583 |
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Expressions et termes fréquents
algebraic Amer bijective proof border strip Catalan numbers character coefficients column combinatorial proof Combinatorial Theory compute Corollary cycle Deduce defined definition denote the number diagonal diagram digraph Discrete Math edges eigenvalues elements entries equal equation equivalent Example Exercise exponential exponential generating function finite formula Gessel given GL(V graph Hence instance integers inversion irreducible jeu de taquin Knuth labeled lattice paths Laurent series Lemma linear Littlewood-Richardson Littlewood-Richardson rule matrix monomial multiset n-set noncommutative noncrossing noncrossing partitions nonnegative Note obtain permutation permutation matrix plane partitions polynomial poset power series proof follows proof of Theorem Proposition rational representation result RSK algorithm satisfying Schur functions Schützenberger Section sequence Show skew SSYT Stanley subset symmetric functions symmetric group SYT of shape tableau unique variables vector vertex set vertices WESn