Introduction. Algebraic combinatorics is concerned with the interaction between combinatorics and such other branches of mathematics as commutative algebra, algebraic geometry, algebraic topology, and representation theory. Many of the major open problems of algebraic combinatorics are related to positivity questions, i.e., showing that certain integers are nonnegative. The significance of positivity to algebraic combinatorics stems from the fact that a nonnegative integer can have both a combinatorial and an algebraic interpretation. The archetypal algebraic interpretation of a nonnegative integer is as the dimension of a vector space. Thus to show that a certain integer m is nonnegative, it suces to nd a vector space Vm of dimension m. Similarly to show that m n, it suces to nd an injective map Vm ! V n or surjective map V n ! Vm . Of course the inequality m n is equivalent to the positivity statement n m 0, while the injectivity of the map ' : Vm ! V n is equivalent to the

In the first part of the paper we survey some far-reaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concerning the simplex algorithm. We describe subexponential randomized pivot rules and upper bounds on the diameter of graphs of polytopes. 1 Introduction: A convex polyhedron is the intersection P of a finite number of closed halfspaces in R d . P is a d-dimensional polyhedron (briefly, a d-polyhedron) if the points in P affinely span R d . A convex d-dimensional polytope (briefly, a d-polytope) is a bounded convex d-polyhedron. Alternatively, a convex d-polytope is the convex hull of a finite set of points which affinely span R d . A (nontrivial) face F of a d-polyhedron P is the intersection of P with a supporting hyperplane. F itself is a polyhedron of some lower dimension. If the dimension of F is k we call F a k-face of P . The empty set and P itself are...

Abstract. This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of M-shellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the rank-numbers of M-shellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex ear-decomposition, and, using results of Kalai and Stanley on h-vectors of simplicial polytopes, we show that h-vectors of pure rank-d simplicial complexes that have this property satisfy h0 ≤ h1 ≤ ·· · ≤ h [d/2] and hi ≤ hd−i for 0 ≤ i ≤ [d/2]. We then show that the abstract simplicial complex formed by the collection of independent sets of a matroid (or matroid complex) admits a special type of convex eardecomposition called a PS ear-decomposition. This enables us to construct an associated M-shellable poset, whose set of rank-numbers is the h-vector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi [17] that the h-vector of a matroid complex satisfies the above two sets of inequalities. 1.

. We start with a theorem of Perles on the k-skeleton, Skel k (P ) (faces of dimension k) of d- polytopes P with d+b vertices for large d. The theorem says that for fixed b and d, if d is sufficiently large, then Skel k (P ) is the k-skeleton of a pyramid over a (d \Gamma 1)-dimensional polytope. Therefore the number of combinatorially distinct k-skeleta of d-polytopes with d + b vertices is bounded by a function of k and b alone. Next we replace b (the number of vertices minus the dimension) by related but deeper invariants of P , the g-numbers. For a d-polytope P there are [d=2] invariants g1 (P ); g2 (P ); :::; g [d=2] (P ) which are of great importance in the combinatorial theory of polytopes. We study polytopes for which g k is small and carried away to related and slightly related problems. Key words: Convex polytopes, skeleton, simplicial sphere, simplicial manifold, f-vector, g- theorem, ranked atomic lattices, stress, rigidity, sunflower, lower bound theorem, elementary poly...

this paper is to present mathematical ideas and

We discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix. We also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron.

Let I ⊂ R be a graded ideal in the polynomial ring R = K[x1,..., xn] where K is a field, and fix a term order <. It has been shown in [17] that the Hilbert functions of the local cohomology modules of R/I are bounded by those of R / in(I), where in(I) denotes the initial ideal of I with respect to <. In this note we study the question

A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, ∆lex — an operation that transforms a monomial ideal of S = k[xi: i ∈ N] that is finitely generated in each degree into a squarefree strongly stable ideal — is defined and studied. It is proved that (in contrast to the reverse lexicographic case) a squarefree strongly stable ideal I ⊂ S is fixed by lexicographic shifting if and only if I is a universal squarefree lexsegment ideal (abbreviated USLI) of S. Moreover, in the case when I is finitely generated and is not a USLI, it is verified that all the ideals in the sequence {∆i lex (I)} ∞ i=0 are distinct. The limit ideal ∆(I) = limi→ ∞ ∆i lex (I) is well defined and is a USLI that depends only on a certain analog of the Hilbert function of I.

We try to understand the behavior of algebraic shifting with respect to some basic constructions on simplicial complexes, such as union, coning, and (more generally) join. In particular, for the disjoint union of simplicial complexes we prove ∆(K ˙∪L) = ∆(∆(K) ˙∪∆(L)) (conjectured by Kalai [2]), and for the join we give an example of simplicial complexes K and L for which ∆(K ∗ L) ̸ = ∆(∆(K) ∗ ∆(L)) (disproving a conjecture by Kalai [2]), where ∆ denotes the (exterior) algebraic shifting operator. We develop a ’homological ’ point of view on algebraic shifting which is used throughout this work. 1

In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for the face numbers of convex polytopes and its many extensions is the second topic. Next are general properties of subsets of the vertices of the discrete n-dimensional cube and some relations with questions of extremal and probabilistic combinatorics. Our fourth topic is tree enumeration and random spanning trees, and finally, some combinatorial and geometrical aspects of the simplex method for linear programming are considered.