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1996 | Buch

Background Kapitel lesen Erstes Kapitel lesen

Combinatorics and Commutative Algebra

verfasst von: Richard P. Stanley

Verlag: Birkhäuser Boston

Buchreihe : Progress in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

Some remarkable connections between commutative algebra and combinatorics have been discovered in recent years. This book provides an overview of two of the main topics in this area. The first concerns the solutions of linear equations in nonnegative 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 topology broadens access to this material for non-specialists.

New to this edition is a chapter surveying more recent work related to face rings, focusing on applications to f-vectors.

Inhaltsverzeichnis

Frontmatter
Chapter 0. Background
Abstract
The purpose of this introduction is to provide the reader with the relevant background from combinatorics, algebra, and topology for understanding of the text. In general the reader may prefer to begin with Chapter 1 and refer back to this chapter only when necessary. We assume the reader is familiar with standard (first-year graduate) material but has no specialized knowledge of combinatorics, commutative algebra, homological algebra, or algebraic topology.
Chapter I. Nonnegative Integral Solutions to Linear Equations
Abstract
The first topic will concern the problem of solving linear equations in nonnegative integers. In particular, we will consider the following problem which goes back to MacMahon. Let Hn(r) := number of n × n ℕ-matrices having line sums r, where a line is a row or column, and an ℕ-matrix is a matrix whose entries belong to ℕ. Such a matrix is called an integer stochastic matrix or magic square. Keeping r fixed, one finds that Hn(0) = 1, Hn(1) = n!, and Anand, Dumir and Gupta [5] showed that
$$ \sum\limits_{n\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ > } 0} {\frac{{H_n \left( 2 \right)x^n }} {{\left( {n!} \right)^2 }} = \frac{{e^{x/2} }} {{\sqrt {1 - x} \cdot }}} $$
See also Stanley [154, Ex. 6.11]. Keeping n fixed, one finds that H1(r) 1,H2(r) = r + 1, and MacMahon [119, Sect. 407] showed that
$$ H_3 \left( r \right) = \left( {\begin{array}{*{20}c} {r + 4} \\ 4 \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {r + 3} \\ 4 \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {r + 2} \\ 4 \\ \end{array} } \right) \cdot $$
Guided by this evidence Anand, Dumir and Gupta [5] formulated the following
Chapter II. The Face Ring of a Simplicial Complex
Abstract
Let Δ be a finite simplicial complex on the vertex set V = {x1,..., xn}. Recall that this means that Δ is a collection of subsets of V such that \( F \subseteq G \in \Delta \Rightarrow F \in \Delta {\mathbf{ }}{\text{and}}{\mathbf{ }}{\text{\{ }}x_1 {\text{\} }} \in \Delta {\mathbf{ }}{\text{for}}{\mathbf{ }}{\text{all}}{\mathbf{ }}x_i \in V. \) . The elements of Δ are called faces. If F ∈ Δ, then define dim F: = |F| − 1 and dim Δ: = maxF∈Δ(dim F). Let d = dim Δ + 1. Given any field k we now define the face ring (or Stanley-Reisner ring) k[Δ] of the complex Δ.
Chapter III. Further Aspects of Face Rings
Abstract
In this chapter we will briefly survey some additional topics related to combinatorics and commutative algebra, mostly dealing with the face ring of a simplicial complex. Our main focus will be on properties of face rings which have applications to combinatorics.
Backmatter
Metadaten
Titel
Combinatorics and Commutative Algebra
verfasst von
Richard P. Stanley
Copyright-Jahr
1996
Verlag
Birkhäuser Boston
Electronic ISBN
978-0-8176-4433-8
Print ISBN
978-0-8176-4369-0
DOI
https://doi.org/10.1007/b139094