Paul H. Edelman

Meet-distributive lattices and the anti-exchange closure

This paper defines the anti-exchange closure, a generalization of the order ideals of a partially ordered set. Various theorems are proved about this closure. The main theorem presented is that a latticeL is the lattice of closed sets of an anti-exchange closure if and only if it is a meet-distributive lattice. This result is used to give a combinatorial interpretation of the zetapolynomial of a meet-distributive lattice.

Chain enumeration and non-crossing partitions

A formula is established for the number of chains with designated ranks in the non-crossing partition lattice. As corollaries, certain results of Kreweras are obtained. Non-crossing partitions are then generalized in two ways, and similar problems are solved.

Hyperplane arrangements with a lattice of regions

A hyperplane arrangement is a finite set of hyperplanes through the origin in a finite-dimensional real vector space. Such an arrangement divides the vector space into a finite set of regions. Every such region determines a partial order on the set of all regions in which these are ordered according to their combinatorial distance from the fixed base region.We show that the base region is simplicial whenever the poset of regions is a lattice and that conversely this condition is sufficient for the lattice property for three-dimensional arrangements, but not in higher dimensions. For simplicial arrangements, the poset of regions is always a lattice.In the case of supersolvable arrangements (arrangements for which the lattice of intersections of hyperplanes is supersolvable), the poset of regions is a lattice if the base region is suitably chosen. We describe the geometric structure of such arrangements and derive an expression for the rank-generating function similar to a known one for Coxeter arrangements. For arrangements with a lattice of regions we give a geometric interpretation of the lattice property in terms of a closure operator defined on the set of hyperplanes.The results generalize to oriented matroids. We show that the adjacency graph (and poset of regions) of an arrangement determines the associated oriented matroid and hence in particular the lattice of intersections.

The Shapley value on convex geometries

Abstract A game on a convex geometry is a real-valued function defined on the family L of the closed sets of a closure operator which satisfies the finite Minkowski–Krein–Milman property. If L is the boolean algebra 2 N then we obtain an n -person cooperative game. Faigle and Kern investigated games where L is the distributive lattice of the order ideals of the poset of players. We obtain two classes of axioms that give rise to a unique Shapley value for games on convex geometries.

Free arrangements and rhombic tilings

AbstractLet Z be a centrally symmetric polygon with integer side lengths. We answer the following two questions:(1)When is the associated discriminantal hyperplane arrangementfree in the sense of Saito and Terao?(2)When areall of the tilings of Z by unit rhombicoherent in the sense of Billera and Sturmfels? Surprisingly, the answers to these two questions are very similar. Furthermore, by means of an old result of MacMahon on plane partitions and some new results of Elnitsky on rhombic tilings, the answer to the first question helps to answer the second. These results then also give rise to some interesting geometric corollaries. Consideration of the discriminantal arrangements for some particular octagons leads to a previously announced counterexample to the conjecture by Saito [ER2] that the complexified complement of a real free arrangement is aK (π, 1) space.

A note on voting

Abstract A framework is presented in which to analyze the power of players in a cooperative game in which only certain coalitions are allowed. The allowable coalitions are characterized by a closure operator that combinatorially abstracts the notion of convexity. The Shapley-Shubik index is extended to this situation, allowing a computation of the power of each player. One example of this framework, voting on a one-dimensional spectrum, is analyzed and it is shown that the powerful players are mid-way between the middle and the extremes. This framework is also applied to an empirical study of power of the Supreme Court. Implications and generalizations are also discussed.

Paradoxes of Fair Division

Two or more players are required to divide up a set of indivisible items that they can rank from best to worst. They may, as well, be able to indicate preferences over subsets, or packages, of items. The main criteria used to assess the fairness of a division are efficiency (Pareto-optimality) and envy-freeness. Other criteria are also suggested, including a Rawlsian criterion that the worst-off player be made as well off as possible and a scoring procedure, based on the Borda count, that helps to render allocations as equal as possible. Eight paradoxes, all of which involve unexpected conflicts among the criteria, are described and classified into three categories, reflecting (1) incompatibilities between efficiency and envy-freeness, (2) the failure of a unique efficient and envy-free division to satisfy other criteria, and (3) the desirability, on occasion, of dividing up items unequally. While troublesome, the paradoxes also indicate opportunities for achieving fair division, which will depend on the fairness criteria one deems important and the trade-offs one considers acceptable.

The Bruhat order of the symmetric group is lexicographically shellable

In this note we present an elementary proof that the Bruhat order of the symmetric group S, is lexicographically shellable and hence Cohen-Macaulay. Using a theorem of Verma we obtain as a corollary that LA(Sn), the simplicial complex of chains of S, is a double cone over a triangulation of a sphere of dimension () 2. We will employ the notation and terminology of Bjorner [2]. A finite poset P is said to be bounded if it has a maximum and a minimum element, denoted I and 0 respectively. It is calledpure if all of its maximal chains are the same length and it is graded if it is both bounded and pure. The rank of P is the length of a maximal chain. An element x of a graded poset P has a well-defined rank p(x) equal to the length of an unrefinable chain from 0 to x in P. If P is bounded let P be the poset P {0, 1}. The order complex /\(P) of a poset P is the simplicial complex of all chains in P. A poset is said to be shellable if /\(P) is shellable. For the definition of a shellable complex see [2] or [4]. Similarly P is called Cohen-Macaulay if /\(P) is. See [1], [2] or [6] for the definition and significance of a Cohen-Macaulay complex. Let C(P) be the set of covering relations

Fair division of indivisible items among people with similar preferences

Abstract We study divisions of a set of indivisible items among three or more people who have the same strict preferences on items but can have different preferences on subsets of items. Preferences on subsets are assumed representable by additive utilities. Each item is received by exactly one person and no payments are involved. The paper focuses on envy-freeness within a division and Pareto optimality among divisions. We characterize envy-free divisions through a notion of convex dominance and observe that a situation can have envy-free divisions none of which is Pareto-optimal.

Chains in the lattice of noncrossing partitions

Abstract The lattice of noncrossing set partitions is known to admit an R -labeling. Under this labeling, maximal chains give rise to permutations. We discuss structural and enumerative properties of the lattice of noncrossing partitions, which pertain to a new permutation statistic, m (σ), defined as the number of maximal chains labeled by σ. Mobius inversion results and related facts about the lattice of unrestricted set partitions are also presented.