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

We prove that the cd-index of a convex polytope satisfies a strong monotonicity property with respect to the cd-indices of any face and its link. As a consequence, we prove for d-dimensional polytopes a conjecture of Stanley that the cd-index is minimized on the d- dimensional simplex. Moreover, we prove the upper bound theorem for the cd-index, namely that the cd-index of any d-dimensional polytope with n vertices is at most that of C(n; d), the d-dimensional cyclic polytope with n vertices.

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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.

Abstract. Poset-theoretic generalizations of set-theoretic committee constructions are presented. The structure of the corresponding subposets is described. Sequences of irreducible fractions associated to the principal order ideals of finite bounded posets are considered and those related to the Boolean lattices are explored; it is shown that such sequences inherit all the familiar properties of the Farey sequences. 1. Introduction and

The problem of finding a triangulation of a convex three-dimensional polytope with few tetrahedra is NP-hard. We discuss other related complexity results.

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(without boundary) is with no doubt one of the most prominent tasks in topology ever since Poincaré’s fundamental work [88] on ≪l’analysis situs≫ appeared in 1904. There are various ways for constructing 3-manifolds, some of which that are general enough to yield all 3-manifolds (orientable or nonorientable) and some that produce only particular types or classes of examples. According to Moise [73], all 3-manifolds can be triangulated. This implies that there are only countably many distinct combinatorial (and therefore at most so many different topological) types that result from gluing together tetrahedra. Another way to obtain 3-manifolds is by starting with a solid 3-dimensional polyhedron for which surface faces are pairwise identified (see, e.g., Seifert [98] and Weber and Seifert [118]). Both approaches are rather general and, on the first sight, do not give much control on the kind of manifold we can expect as an outcome. However, if we want to determine the topological type of some given triangulated 3-manifold, then small or minimal triangulations

Abstract. Given integers k 1 and n 0, there is a unique way of writing n as n = nk k + nk 1 n1 k 1 1 so that 0 n1 < < nk 1 < nk. Using this representation, the Kruskal-Macaulay function of n is de…ned as @ k (n) = nk 1 k 1 + nk 1 1 k 2 n1 1

We define Dumont's statistic on the symmetric group S n to be the function dmc: S n ! N which maps a permutation oe to the number of distinct nonzero letters in code(oe). Dumont showed that this statistic is Eulerian. Naturally extending Dumont's statistic to the rearrangement classes of arbitrary words, we create a generalized statistic which is again Eulerian. As a consequence, we show that for each distributive lattice J(P ) which is a product of chains, there is a poset Q such that the f-vector of Q is the h-vector of J(P ). This strengthens for products of chains a result of Stanley concerning the flag h-vectors of Cohen-Macaulay complexes. We conjecture that the result holds for all finite distributive lattices. 1