good, does not support this; the label ‘Example’ is given in a rather small font followed by a ‘PROOF,’ and the body of an example is nonitalic, utterly unlike other statements accompanied by demonstrations. The proofs of some theorems are continued after the statements and proofs of needed lemmas with no helpful demarcations (“Continuation of proof of Theorem . . . ”). Finally, at least one lemma (3.34) has a corollary with no theorem or proposition in sight; the status of Lemma 3.36 is also unclear. Lemma 1.16 on page 12 giving a closed formula for Stirling numbers of the second kind is important enough to figure in the proof of capstone Theorem 3.71 (Schlömilch’s formula) on page 114. This nonstandard approach to chapter organization may impede some readers’ comprehending the structure of the author’s highly purposeful discussion. Review of 3 Enumerative Combinatorics Author of Book: Charalambos A. Charalambides Publisher: Chapman & Hall/CRC ISBN L-58488-290-5, Hard Cover, 609 pages

Enumerative Combinatorics

polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook.

polymake: a Framework for Analyzing Convex Polytopes

Introduction Polytopes Topology and combinatorics of simplicial complexes Commutative and homological algebra of simplicial complexes Cubical complexes Toric and quasitoric manifolds Moment-angle complexes Cohomology of moment-angle complexes and combinatorics of triangulated manifolds Cohomology rings of subspace arrangement complements Bibliography Index.

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Torus Actions and their Applications in Topology and Combinatorics

These lecture notes for the IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004) provide an overview of poset topology. These notes include introductory material, as well as recent developments and open problems. Some of the topics covered are: subspace arrangements, graph complexes, group actions on poset homology, shellability, recursive techniques, and fiber theorems.

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Poset Topology: Tools and Applications

A convex polytope P can be specified in two ways: as the convex hull of the vertex set V of P , or as the intersection of the set H of its facet-inducing halfspaces. The vertex enumeration problem is to compute V from H>. The facet enumeration problem is to compute H from V. These two problems are essentially equivalent under point/hyperplane duality. They are among the central computational problems in the theory of polytopes. It is open whether they can be solved in time polynomial in | H | + | V | and the dimension. In this paper we consider the main known classes of algorithms for solving these problems. We argue that they all have at least one of two weaknesses: inability to deal well with “degeneracies”, or, inability to control the sizes of intermediate results. We then introduce families of polytopes that exercise those weaknesses. Roughly speaking, fat-lattice or intricate polytopes cause algorithms with bad degeneracy handling to perform badly; dwarfed polytopes cause algorithms with bad intermediate size control to perform badly. We also present computational experience with trying to solve these problem on these hard polytopes, using various implementations of the main algorithms.

How good are convex hull algorithms

This paper generalizes the Dehn-Sommerville equations for simplicial spheres to related classes of objects. The underlying motivation is to understand the combinatorial structure of arbitrary polytopes, that is, polytopes that are not necessarily simplicial. Towards this end we determine the affine span of the extended f-vectors of d-polytopes. A polytope is the convex hull of finitely many points in R e . We will generally consider a polytope of affine dimension d to be a subset of Re; this is referred to as a d-polytope. A face of a polytope is the intersection of a supporting hyperplane with the polytope. For the most part we identify a polytope P with the abstract cell complex (or lattice) realized by the boundary of P, and write a face of P as the set of vertices of P it contains. That is, we will shorten F = c o n v { v 0 . . . . . Vk} to F={Vo,.. . ,Vk} when there is no risk of confusion. By convention, the empty set is considered a ( -1)-dimensional face, and the polytope itself is a d-dimentional face; these faces will be called improper faces of P. A polytope P is called simplicial if each of its faces, except possibly P itself, is a simplex (the convex hull of affinely independent points). We will write Y (respectively, ~ ) for the set of all (respectively, all simplicial) d-polytopes. The number of/-dimensional faces (or/-faces) of a polytope P is written f , and f(P)=(fo,fl . . . . . fd-1) is called the f-vector of P. The set off -vectors of all (simplicial) polytopes is written f ( ~ e ) (f(~d)). A certain transformation on the f-vectors of simplicial polytopes has arisen in a number of different contexts, and will play an important part here. For a d-polytope P define the h-vector h(P)=(ho, hl,...,he) b y h i = i ( 1 ) i s ( d j ) s= o d i f j 1 (here we use the convention

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Generalized Dehn-Sommerville Relations for Polytopes, Spheres and Eulerian Partially Ordered Sets

A new index for convex polytopes is introduced. It is a vector whose length is the dimension of the linear span of the flag vectors of polytopes. The existence of this index is equivalent to the generalized Dehn-Sommerville equations. It can be computed via a shelling of the polytope. The ranks of the middle perversity intersection homology of the associated toric variety are computed from the index. This gives a proof of a result of Kalai on the relationship between the Betti numbers of a polytope and those of its dual.

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A new index for polytopes

This chapter discusses some techniques that have been successful in analyzing the combinatorial aspects of convex polytopes. A convex polyhedron is a subset of ℝ d that is the intersection of a finite number of closed half-spaces. A bounded convex polyhedron is called a convex polytope. The following can be regarded as the fundamental theorem of convex polytopes: P ⊂ ℝ d is a polytope if and only if it is the convex hull of a finite set of points in ℝ d . This theorem and related results are foundational to the theory of linear programming duality, and one of the central themes of combinatorial optimization is to make this conversion for special polytopes related to specific programming problems. The problem of developing algorithms to convert from one description of a polytope to the other arises in mathematical programming and computational geometry. The second theorem states that the collection of all the faces of a polyhedron P , ordered by inclusion, is a lattice. This lattice is called the face lattice or boundary complex of P , and two polytopes are (combinatorially) equivalent if their face lattices are isomorphic. The third theorem states that the face lattices of P and P * are anti-isomorphic. Two polytopes with anti-isomorphic face lattices are said to be dual. Two important dual classes of d -polytopes are the class of simplicial d -polytopes and the class of simple d -polytopes.

Combinatorial Aspects of Convex Polytopes

The extended f -vectors of 4-polytopes

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Margaret M. Bayer

Papers

Generalized Dehn-Sommerville Relations for Polytopes, Spheres and Eulerian Partially Ordered Sets

A new index for polytopes

Combinatorial Aspects of Convex Polytopes

The extended f -vectors of 4-polytopes

Counting Faces and Chains in Polytopes and Posets