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 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. 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.