1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

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Convex bodies : the Brunn-Minkowski theory

Preface Introduction Conventions and terminology 1. Commutative rings and modules 2. prime ideals 3. Properties of extension rings 4. Valuation rings 5. Dimension theory 6. Regular sequences 7. Regular rings 8. Flatness revisited 9. Derivations 10. I-smoothness 11. Applications of complete local rings Appendices References Index.

Commutative Ring Theory

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.

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A Dynamic Survey of Graph Labeling

This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.

Spectra of graphs

Richard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This thoroughly revised second edition of Volume 1 includes ten new sections and more than 300 new exercises, most with solutions, reflecting numerous new developments since the publication of the first edition in 1986. The material in Volume 1 was chosen to cover those parts of enumerative combinatorics of greatest applicability and with the most important connections with other areas of mathematics. The four chapters are devoted to an introduction to enumeration (suitable for advanced undergraduates), sieve methods, partially ordered sets, and rational generating functions. Much of the material is related to generating functions, a fundamental tool in enumerative combinatorics. In this new edition, the author brings the coverage up to date and includes a wide variety of additional applications and examples, as well as updated and expanded chapter bibliographies. Many of the less difficult new exercises have no solutions so that they can more easily be assigned to students. The material on P-partitions has been rearranged and generalized; the treatment of permutation statistics has been greatly enlarged; and there are also new sections on q-analogues of permutations, hyperplane arrangements, the cd-index, promotion and evacuation, and differential posets.

Enumerative Combinatorics: Volume 1

Enumerative Combinatorics: Index

This text offers an overview of two of the main topics in the connections between commutative algebra and combinatorics. The first concerns the solutions of linear equations in non-negative integers. Applications are given to the enumeration of integer stochastic matrices (or magic squares), the volume of polytopes, combinatorial reciprocity theorems and related results. The second topic deals with the face ring of a simplicial complex, and includes a proof of the upper bound conjecture for spheres. An introductory chapter giving background information in algebra, combinatorics and toplogy aims to broaden access to this material for non-specialists. This edition contains a chapter surveying more recent work related to face rings, focusing on applications to f-vectors. Also included is information on subcomplexes and subdivisions of simplicial complexes, and an application to spline theory.

Combinatorics and commutative algebra

http://www-math.mit.edu/~rstan/pubs/pubfiles/30.pdf

Hilbert functions of graded algebras

A sequence a,, a,, . . . , a, of real numbers is said to be unimodal if for some 0 s j _c n we have a, 5 a , 5 . . 5 ai 2 a,,, 2 . . 2 a,, and is said to be logarithmically concave (or log-concave for short) if a: 2 a, ,a ,+ , for all 1 5 i 5 n 1. Clearly a log-concave sequence of positive terms is unimodal. Let us say that the sequence a,, a,, . . . , a, has no internal zeros if there do not exist integers 0 5 i < j < k 5 n satisfying a, f 0, a, = 0, ak # 0. Then in fact a nonnegative log-concave sequence with no internal zeros is unimodal. The sequence a,, a , , . . . , a, is called symmetric if a, = a,-, for 0 5 i 5 n. Thus a symmetric unimodal sequence a,, a, , . . . , a,, has its maximum at the middle term ( n even) or middle two terms (n odd). We also say that a polynomial a, + a,4 +. . + an$ has a certain property (such as unimodal, log-concave, or symmetric) if its sequence a,, a,, . . . , a, of coefficients has that property. Our object here is to survey the surprisingly rich variety of methods for showing that a sequence is log-concave or unimodal. For each method we will give examples of its applicability to combinatorially defined sequences that arise naturally from problems in algebra, combinatorics, and geometry. We make no attempt, however, to give a comprehensive account of all work done in this area.

Richard P. Stanley

Papers

Enumerative Combinatorics: Volume 1

Enumerative Combinatorics: Index

Combinatorics and commutative algebra

Hilbert functions of graded algebras

Log‐Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry