**Graduate Studies in Mathematics**

Volume: 195;
2018;
308 pp;
Hardcover

MSC: Primary 05; 11; 52; 68;

**Print ISBN: 978-1-4704-2200-4
Product Code: GSM/195**

List Price: $73.00

AMS Member Price: $58.40

MAA Member Price: $65.70

**Electronic ISBN: 978-1-4704-4996-4
Product Code: GSM/195.E**

List Price: $73.00

AMS Member Price: $58.40

MAA Member Price: $65.70

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#### Supplemental Materials

# Combinatorial Reciprocity Theorems: An Invitation to Enumerative Geometric Combinatorics

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*Matthias Beck; Raman Sanyal*

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.

#### Readership

Advanced undergraduate students and graduate students learning combinatorics; instructors teaching such courses.

#### Reviews & Endorsements

All in all, the authors gather numerous topics in a single impressive work and organize the material in a very engaging and unique fashion.

-- Philippe Nadeau, Mathematical Reviews

#### Table of Contents

# Table of Contents

## Combinatorial Reciprocity Theorems: An Invitation to Enumerative Geometric Combinatorics

- Cover Cover11
- Title page i2
- Title page iii4
- Preface ix10
- Chapter 1. Four Polynomials 116
- Chapter 2. Partially Ordered Sets 2944
- Chapter 3. Polyhedral Geometry 5166
- 3.1. Inequalities and Polyhedra 5267
- 3.2. Polytopes, Cones, and Minkowski–Weyl 6075
- 3.3. Faces, Partially Ordered by Inclusion 6681
- 3.4. The Euler Characteristic 7287
- 3.5. Möbius Functions of Face Lattices 8196
- 3.6. Uniqueness of the Euler Characteristics and Zaslavsky’s Theorem 86101
- 3.7. The Brianchon–Gram Relation 91106
- Notes 94109
- Exercises 96111

- Chapter 4. Rational Generating Functions 107122
- 4.1. Matrix Powers and the Calculus of Polynomials 107122
- 4.2. Compositions 115130
- 4.3. Plane Partitions 117132
- 4.4. Restricted Partitions 120135
- 4.5. Quasipolynomials 122137
- 4.6. Integer-point Transforms and Lattice Simplices 124139
- 4.7. Gradings of Cones and Rational Polytopes 129144
- 4.8. Stanley Reciprocity for Simplicial Cones 132147
- 4.9. Chain Partitions and the Dehn–Sommerville Relations 137152
- Notes 143158
- Exercises 145160

- Chapter 5. Subdivisions 155170
- 5.1. Decomposing a Polyhedron 155170
- 5.2. Möbius Functions of Subdivisions 165180
- 5.3. Beneath, Beyond, and Half-open Decompositions 168183
- 5.4. Stanley Reciprocity 174189
- 5.5. ℎ*-vectors and 𝑓-vectors 176191
- 5.6. Self-reciprocal Complexes and Dehn–Sommerville Revisited 181196
- 5.7. A Combinatorial Triangulation 188203
- Notes 193208
- Exercises 195210

- Chapter 6. Partially Ordered Sets, Geometrically 203218
- Chapter 7. Hyperplane Arrangements 235250
- 7.1. Chromatic, Order Polynomials, and Subdivisions Revisited 236251
- 7.2. Flats and Regions of Hyperplane Arrangements 239254
- 7.3. Inside-out Polytopes 245260
- 7.4. Alcoved Polytopes 250265
- 7.5. Zonotopes and Tilings 261276
- 7.6. Graph Flows and Totally Cyclic Orientations 273288
- Notes 280295
- Exercises 281296

- Bibliography 287302
- Notation Index 297312
- Index 301316
- Back Cover Back Cover1325