GRADUATE STUDIES IN MATHEMATICS 195 Combinatorial Reciprocity Theorems An Invitation to Enumerative Geometric Combinatorics Matthias Beck Raman Sanyal 10.1090/gsm/195 Combinatorial Reciprocity Theorems An Invitation to Enumerative Geometric Combinatorics Matthias Beck Raman Sanyal GRADUATE STUDIES IN MATHEMATICS 195 Combinatorial Reciprocity Theorems An Invitation to Enumerative Geometric Combinatorics Matthias Beck Raman Sanyal EDITORIAL COMMITTEE Daniel S. Freed (Chair) Bjorn Poonen Gigliola Staffilani Jeff A. Viaclovsky 2010 Mathematics Subject Classification. Primary 05Axx, 05C31, 05E45, 11P21, 52B05, 52B11, 52B20, 52B45, 52C07, 68R05. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-195 Library of Congress Cataloging-in-Publication Data Names: Beck, Matthias, author. | Sanyal, Raman, 1978-author. Title: Combinatorial reciprocity theorems : an invitation to enumerative geometric combinatorics / Matthias Beck, Raman Sanyal. Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Gradu- ate studies in mathematics ; volume 195 | Includes bibliographical references and index. Identifiers: LCCN 2018034294 | ISBN 9781470422004 (alk. paper) Subjects: LCSH: Combinatorial geometry. | Combinatorial analysis. | AMS: Combinatorics – Enumerative combinatorics – Enumerative combinatorics. msc — Combinatorics – Graph theory – Graph polynomials. msc | Combinatorics – Algebraic combinatorics – Combinatorial aspects of simplicial complexes. msc | Number theory – Additive number theory; partitions – Lattice points in specified regions. msc | Convex and discrete geometry – Polytopes and polyhedra – Combinatorial properties (number of faces, shortest paths, etc.). msc | Convex and discrete geometry – Polytopes and polyhedra – n-dimensional polytopes. msc | Convex and discrete geometry – Polytopes and polyhedra – Lattice polytopes (including relations with commutative algebra and algebraic geometry). msc | Convex and discrete geometry – Polytopes and polyhedra – Dissections and valuations (Hilbert’s third problem, etc.). msc | Convex and discrete geometry – Discrete geometry – Lattices and convex bodies in n dimensions. msc | Computer science – Discrete mathematics in relation to computer science – Combinatorics. msc Classification: LCC QA167.B3545 2018 | DDC 511/.6–dc23 LC record available at https://lccn.loc.gov/2018034294 Copying and reprinting. 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Visit the AMS home page at https://www.ams.org/ 10987654321 232221201918 Contents Preface ix Chapter 1. Four Polynomials 1 §1.1. Graph Colorings 1 §1.2. Flows on Graphs 7 §1.3. Order Polynomials 12 §1.4. Ehrhart Polynomials 15 Notes 21 Exercises 23 Chapter 2. Partially Ordered Sets 29 §2.1. Order Ideals and the Incidence Algebra 29 §2.2. The M¨obius Function and Order Polynomial Reciprocity 33 §2.3. Zeta Polynomials, Distributive Lattices, and Eulerian Posets 36 §2.4. Inclusion–Exclusion and M¨obius Inversion 39 Notes 45 Exercises 46 Chapter 3. Polyhedral Geometry 51 §3.1. Inequalities and Polyhedra 52 §3.2. Polytopes, Cones, and Minkowski–Weyl 60 §3.3. Faces, Partially Ordered by Inclusion 66 §3.4. The Euler Characteristic 72 §3.5. M¨obius Functions of Face Lattices 81 v vi Contents §3.6. Uniqueness of the Euler Characteristics and Zaslavsky’s Theorem 86 §3.7. The Brianchon–Gram Relation 91 Notes 94 Exercises 96 Chapter 4. Rational Generating Functions 107 §4.1. Matrix Powers and the Calculus of Polynomials 107 §4.2. Compositions 115 §4.3. Plane Partitions 117 §4.4. Restricted Partitions 120 §4.5. Quasipolynomials 122 §4.6. Integer-point Transforms and Lattice Simplices 124 §4.7. Gradings of Cones and Rational Polytopes 129 §4.8. Stanley Reciprocity for Simplicial Cones 132 §4.9. Chain Partitions and the Dehn–Sommerville Relations 137 Notes 143 Exercises 145 Chapter 5. Subdivisions 155 §5.1. Decomposing a Polyhedron 155 §5.2. M¨obius Functions of Subdivisions 165 §5.3. Beneath, Beyond, and Half-open Decompositions 168 §5.4. Stanley Reciprocity 174 §5.5. h∗-vectors and f-vectors 176 §5.6. Self-reciprocal Complexes and Dehn–Sommerville Revisited 181 §5.7. A Combinatorial Triangulation 188 Notes 193 Exercises 195 Chapter 6. Partially Ordered Sets, Geometrically 203 §6.1. The Geometry of Order Cones 204 §6.2. Subdivisions, Linear Extensions, and Permutations 210 §6.3. Order Polytopes and Order Polynomials 214 §6.4. The Arithmetic of Order Cones and P -Partitions 220 Notes 229 Exercises 230 Contents vii Chapter 7. Hyperplane Arrangements 235 §7.1. Chromatic, Order Polynomials, and Subdivisions Revisited 236 §7.2. Flats and Regions of Hyperplane Arrangements 239 §7.3. Inside-out Polytopes 245 §7.4. Alcoved Polytopes 250 §7.5. Zonotopes and Tilings 261 §7.6. Graph Flows and Totally Cyclic Orientations 273 Notes 280 Exercises 281 Bibliography 287 Notation Index 297 Index 301 Preface Combinatorics is not a field, it’s an attitude. Anon A combinatorial reciprocity theorem relates two classes of combinatorial ob- jects via their counting functions: consider a class X of combinatorial objects and let f(n) be the function that counts the number of objects in X of size n,wheresize refers to some specific quantity that is naturally associated with the objects in X . Similar to canonization, it requires two miracles for a combinatorial reciprocity to occur: 1. the function f(n) is the restriction of some reasonable function (e.g., a polynomial) to the positive integers, and 2. the evaluation f(−n) is an integer of the same sign σ = ±1 for all n ∈ Z>0. In this situation it is only human to ask if σf(−n) has a combinatorial meaning, that is, if there is a natural class X ◦ of combinatorial objects such that σf(−n) counts the objects of X ◦ of size n (where size again refers to some specific quantity naturally associated to X ◦). Combinatorial reci- procity theorems are among the most charming results in mathematics and, in contrast to canonization, can be found all over enumerative combinatorics and beyond. As a first example we consider the class of maps [k] → Z>0 from the finite set [k]:={1, 2,...,k} into the positive integers, and so f(n)=nk counts the number of maps with codomain [n]. Thus f(n) is the restriction of a polynomial and (−1)kf(−n)=nk satisfies our second requirement above. This relates the number of maps [k] → [n] to itself. This relation is a genuine combinatorial reciprocity but the impression one is left with is that of being ix x Preface underwhelmed rather than charmed. Later in the book it will become clear that this example is not boring at all, but for now let’s try again. The term combinatorial reciprocity theorem was coined by Richard Stan- ley in his 1974 paper [162] of the same title, in which he developed a firm foundation of the subject. Stanley starts with an appealing reciprocity that he attributes to John Riordan: For a set S and d ∈ Z≥0, the collection of 1 d-subsets of S is S := {A ⊆ S : |A| = d} . d For d fixed, the number of d-subsets of S depends only on the cardinality |S|, and the number of d-subsets of an n-set is n 1 (0.0.1) f(n)= = n(n − 1) ···(n − d +2)(n − d +1), d d! which is the restriction of a polynomial in n of degree d. From the factoriza- tion we can read off that (−1)df(−n) is a positive integer for every n>0. More precisely, 1 n + d − 1 (−1)df(−n)= n(n +1)···(n + d − 2)(n + d − 1) = , d! d which is the number of d-multisubsets of an n-set, that is, the number of picking d elements from [n] with repetition but without regard to the order in which the elements are picked. Now this is a combinatorial reciprocity! In formulas it reads −n n + d − 1 (0.0.2) (−1)d = . d d This is enticing in more than one way. The identity presents an intriguing connection between subsets and multisubsets via their counting functions, and its formal justification is completely within the realms of an undergradu- ate class in combinatorics. Equation (0.0.2) can be found in Riordan’s book [143] on combinatorial analysis without further comment and, charmingly, Stanley states that his paper [162] can be considered as “further comment”. That further comment is necessary is apparent from the fact that the for- mal proof above falls short of explaining why these two sorts of objects are related by a combinatorial reciprocity. In particular, comparing coefficients in (0.0.2) cannot be the method of choice for establishing more general reci- procity relations. In this book we develop tools and techniques for handling combinatorial reciprocities. However, our own perspective is firmly rooted in geometric combinatorics and, thus, our emphasis is on the geometric nature of the 1All our definitions will look like that: incorporated into the text but bold-faced and so hopefully clearly visible. Preface xi combinatorial reciprocities. That is, for every class of combinatorial objects we associate a geometric object (such as a polytope or a polyhedral complex) in such a way that combinatorial features, including counting functions and reciprocity, are reflected in the geometry.