Abstract. 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 Stanley's 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.
We analyze 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 let 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.
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