Anders Björner

Shellable and Cohen-Macaulay partially ordered sets

In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to R. Stanley, is generalized to posets and shown to imply shellability, while Stanleys main theorem on the Jordan-Holder sequences of such labelings remains valid. Further, we show a number of ways in which shellable posets can be constructed from other shellable posets and complexes. These results give rise to several new examples of CohenMacaulay posets. For instance, the lattice of subgroups of a finite group G is Cohen-Macaulay (in fact shellable) if and only if G is supersolvable. Finally, it is shown that all the higher order complexes of a finite planar distributive lattice are shellable. Introduction. A pure finite simplicial complex A is said to be shellable if its maximal faces can be ordered F,, F2, . . ., Fn in such a way that Fk n ( U *j/ Fj) is a nonempty union of maximal proper faces of Fk for k = 2, 3, . . ., n. It is known that a shellable complex A must be Cohen-Macaulay, that is, a certain commutative ring associated with A is a Cohen-Macaulay ring (see the appendix for details). The notion of shellability, which originated in polyhedral theory, is emerging as a useful concept also in combinatorics with applications in matroid theory and order theory. In this paper we study shellable posets (partially ordered sets), that is, finite posets for which the order complex consisting of all chains x, < x2 < • • • < xk is shellable. The material is organized as follows. After some preliminary remarks in §1, we discuss in §2 a certain type of labeling of the edges of the Hasse diagram of finite posets. We call posets which admit such labeling lexicographically shellable, and we prove that lexicographically shellable posets are indeed shellable. In lexicographically shellable posets the Möbius function can be interpreted as counting certain distinctly labeled maximal chains. We elaborate somewhat on this principle, point out its natural connection with shellability, and exemplify its use. Received by the editors March 12, 1979. Presented to the Society, March 22, 1978, at the Symposium on Relations Between Combinatorics and Other Parts of Mathematics held at Ohio State University. AMS (MOS) subject classifications (1970). Primary 06A10, 06A20; Secondary 05B35, 20D30, 52A25, 57C05.

SHELLABLE NONPURE COMPLEXES AND POSETS. II

This is a direct continuation of Shellable Nonpure Complexes and Posets. I, which appeared in Transactions of the American Mathematical Society 348 (1996), 1299-1327. 8. Interval-generated lattices and dominance order In this section and the following one we will continue exemplifying the applicability of lexicographic shellability to nonpure posets. Let F = {I1, I2, . . . , In} be a family of intervals of integers, by which is meant sets of the form [a, b] = {a, a + 1, . . . , b}, a ≤ b. We assume that there are no containments among these intervals, and that they are ordered so that their left and right endpoints are increasing. Let L(F) be the lattice of all sets that are unions of subfamilies of F , ordered by inclusion. Such interval-generated lattices L(F) were introduced and studied by Greene [G]. Define an edge-labeling λ of L(F) as follows. If A → B is a covering and a = max(B \A), then λ(A→ B) = { −a, if (a+ 1) ∈ A and a is the left endpoint of some I ∈ F ,

On lexicographically shellable posets

Lexicographically shellable partially ordered sets are studied. A new recursive formulation of CL-shellability is introduced and exploited. It is shown that face lattices of convex polytopes, totally semimodular posets, posets of injective and normal words and lattices of bilinear forms are CL-shellable. Finally, it is shown that several common operations on graded posets preserve shellability and CL-shellability.

Chip-firing games on graphs

We analyse the following (solitaire) game: each node of a graph contains a pile of chips, and a move consists of selecting a node with at least as many chips on it as its degree, and letting it send one chip to each of its neighbors. The game terminates if there is no such node. We show that the finiteness of the game and the terminating configuration are independent of the moves made. If the number of chips is less than the number of edges, the game is always finite. If the number of chips is at least the number of edges, the game can be infinite for an appropriately chosen initial configuration. If the number of chips is more than twice the number of edges minus the number of nodes, then the game is always infinite. The independence of the finiteness and the terminating position follows from simple but powerful ‘exchange properties’ of the sequences of legal moves, and from some general results on ‘antimatroids with repetition’, i.e. languages having these exchange properties. We relate the number of steps in a finite game to the least positive eigenvalue of the Laplace operator of the graph.

An Introduction to Cohen-Macaulay Partially Ordered Sets

Combinatorics, algebra and topology come together in a most remarkable way in the theory of Cohen-Macaulay posets. These lectures will provide an introduction to the subject based on the work of Baclawski, Hochster, Reisner and the present authors (see references).

Matroid Applications: Introduction to Greedoids

Introduction Greedoids were invented around 1980 by B. Korte and L. Lovasz. Originally, the main motivation for proposing this generalization of the matroid concept came from combinatorial optimization. Korte and Lovasz had observed that the optimality of a ‘greedy’ algorithm could in several instances be traced back to an underlying combinatorial structure that was not a matroid – but (as they named it) a ‘greedoid’. In subsequent research greedoids have been shown to be interesting also from various non-algorithmic points of view. The basic distinction between greedoids and matroids is that greedoids are modeled on the algorithmic construction of certain sets, which means that the ordering of elements in a set plays an important role. Viewing such ordered sets as words, and the collection of words as a formal language, we arrive at the general definition of a greedoid as a finite language that is closed under the operation of taking initial substrings and satisfies a matroid-type exchange axiom. It is a pleasant feature that greedoids can also be characterized in terms of set systems (the unordered version), but the language formulation (the ordered version) seems more fundamental. Consider, for instance, the algorithmic construction of a spanning tree in a connected graph. Two simple strategies are: (1) pick one edge at a time, making sure that the current edge does not form a circuit with those already chosen; (2) pick one edge at a time, starting at some given node, so that the current edge connects a visited node with an unvisited node.

Linear decision trees: volume estimates and topological bounds

We describe two methods for estimating the size and depth of decision trees where a linear test is performed at each node. Both methods are applied to the question of deciding, by a linear decision tree, whether given n real numbers, some k of them are equal. We show that the minimum depth of a linear decision tree for this problem is Θ(n log(n/k)). The upper bound is easy; the lower bound can be established for k = O(n1/4−e) by a volume argument; for the whole range, however, our proof is more complicated and it involves the use of some topology as well as the theory of Mobius functions.

Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings

A des groupes de Coxeter et des groupes a BN-paire sont associes certaines structures combinatoires: les complexes de Coxeter et les constructions de Tits. On etudie ces structures du point de vue des complexes de Cohen-Macaulay et des complexes «decorticables»

Fixed points in partially ordered sets

In this paper we present a number of theorems about fixed points of mappings of partially ordered sets. Our approach is based on a discrete form of the HopfLefschetz fixed point theorem and on order-theoretical analogs of topological constructions. However, we show by example that the fixed point theory of partially ordered sets cannot be reduced to topological fixed point theory. Nevertheless, a substantial number of previously known results in this field are not only subsumed under our approach but are also extended and refined. This is particularly true in the finite case where certain qualitative properties of the fixed point sets come within reach which are stronger than that of merely being nonvoid. We also show that, somewhat surprisingly, fixed point theory has applications to the question of the existence of complements in finite lattices. Let P be a poset (partially ordered set). A self-map is a function f:

Permutation statistics and linear extensions of posets

Abstract We consider subsets of the symmetric group for which the inversion index and major index are equally distributed. Our results extend and unify results of MacMahon, Foata, and Schutzenberger, and the authors. The sets of permutations under study here arise as linear extensions of labeled posets, and more generally as order closed subsets of a partial ordering of the symmetric group called the weak order. For naturally labeled posets, we completely characterize. as postorder labeled forests, those posets whose linear extension set is equidistributed. A bijection of Foata which takes major index to inversion index plays a fundamental role in our study of equidistributed classes of permutations. We also explore, here, classes of permutations which are invariant under Foatas bijection.