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E-grāmata: Combinatorial Reciprocity Theorems

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  • Formāts: 308 pages
  • Sērija : Graduate Studies in Mathematics
  • Izdošanas datums: 12-Nov-2018
  • Izdevniecība: American Mathematical Society
  • ISBN-13: 9781470449964
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Combinatorial Reciprocity TheoremsPalielināt
  • Formāts: 308 pages
  • Sērija : Graduate Studies in Mathematics
  • Izdošanas datums: 12-Nov-2018
  • Izdevniecība: American Mathematical Society
  • ISBN-13: 9781470449964
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Combinatorial reciprocity is a very interesting phenomenon, which can be described as follows: A polynomial, whose values at positive integers count combinatorial objects of some sort, may give the number of combinatorial objects of a different sort when evaluated at negative integers (and suitably normalized). Such combinatorial reciprocity theorems occur in connections with graphs, partially ordered sets, polyhedra, and more. Using the combinatorial reciprocity theorems as a leitmotif, this book unfolds central ideas and techniques in enumerative and geometric combinatorics.

Written in a friendly writing style, this is an accessible graduate textbook with almost 300 exercises, numerous illustrations, and pointers to the research literature. Topics include concise introductions to partially ordered sets, polyhedral geometry, and rational generating functions, followed by highly original chapters on subdivisions, geometric realizations of partially ordered sets, and hyperplane arrangements.
Preface ix
Chapter 1 Four Polynomials
1(28)
§1.1 Graph Colorings
1(6)
§1.2 Flows on Graphs
7(5)
§1.3 Order Polynomials
12(3)
§1.4 Ehrhart Polynomials
15(14)
Notes
21(2)
Exercises
23(6)
Chapter 2 Partially Ordered Sets
29(22)
§2.1 Order Ideals and the Incidence Algebra
29(4)
§2.2 The Mobius Function and Order Polynomial Reciprocity
33(3)
§2.3 Zeta Polynomials, Distributive Lattices, and Eulerian Posets
36(3)
§2.4 Inclusion-Exclusion and Mobius Inversion
39(12)
Notes
45(1)
Exercises
46(5)
Chapter 3 Polyhedral Geometry
51(56)
§3.1 Inequalities and Polyhedra
52(8)
§3.2 Polytopes, Cones, and Minkowski-Weyl
60(6)
§3.3 Faces, Partially Ordered by Inclusion
66(6)
§3.4 The Euler Characteristic
72(9)
§3.5 Mobius Functions of Face Lattices
81(5)
§3.6 Uniqueness of the Euler Characteristics and Zaslavsky's Theorem
86(5)
§3.7 The Brianchon--Gram Relation
91(16)
Notes
94(2)
Exercises
96(11)
Chapter 4 Rational Generating Functions
107(48)
§4.1 Matrix Powers and the Calculus of Polynomials
107(8)
§4.2 Compositions
115(2)
§4.3 Plane Partitions
117(3)
§4.4 Restricted Partitions
120(2)
§4.5 Quasipolynomials
122(2)
§4.6 Integer-point Transforms and Lattice Simplices
124(5)
§4.7 Gradings of Cones and Rational Polytopes
129(3)
§4.8 Stanley Reciprocity for Simplicial Cones
132(5)
§4.9 Chain Partitions and the Dehn--Sommerville Relations
137(18)
Notes
143(2)
Exercises
145(10)
Chapter 5 Subdivisions
155(48)
§5.1 Decomposing a Polyhedron
155(10)
§5.2 Mobius Functions of Subdivisions
165(3)
§5.3 Beneath, Beyond, and Half-open Decompositions
168(6)
§5.4 Stanley Reciprocity
174(2)
§5.5 h*x-vectors and ƒ-vectors
176(5)
§5.6 Self-reciprocal Complexes and Dehn-Sommerville Revisited
181(7)
§5.7 A Combinatorial Triangulation
188(15)
Notes
193(2)
Exercises
195(8)
Chapter 6 Partially Ordered Sets, Geometrically
203(32)
§6.1 The Geometry of Order Cones
204(6)
§6.2 Subdivisions, Linear Extensions, and Permutations
210(4)
§6.3 Order Polytopes and Order Polynomials
214(6)
§6.4 The Arithmetic of Order Cones and P-Partitions
220(15)
Notes
229(1)
Exercises
230(5)
Chapter 7 Hyperplane Arrangements
235(52)
§7.1 Chromatic, Order Polynomials, and Subdivisions Revisited
236(3)
§7.2 Flats and Regions of Hyperplane Arrangements
239(6)
§7.3 Inside-out Polytopes
245(5)
§7.4 Alcoved Polytopes
250(11)
§7.5 Zonotopes and Tilings
261(12)
§7.6 Graph Flows and Totally Cyclic Orientations
273(14)
Notes
280(1)
Exercises
281(6)
Bibliography 287(10)
Notation Index 297(4)
Index 301
Matthias Beck, San Francisco State University, CA.

Raman Sanyal, Goethe-Universitat Frankfurt, Germany.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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