Abstract:

Paul Edelman and Vic Reiner introduced the concept of an $h$-shelling of a simplicial complex $\Sigma$, which is a shelling in which the family consisting of the minimal faces from the shelling steps forms a simplicial complex. The latter complex is the $h$-complex of $\Sigma$ with respect to the given shelling. For any naturally labeled poset $P$, we consider the $h$-complex $\Delta(P)$ with respect to a certain shelling of the order complex of the lattice of order ideals in $P$; the complex $\Delta(P)$ was introduced by Richard Stanley. The $d$-dimensional faces of $\Delta(P)$ are in bijection with linear extensions of $P$ with $d+1$ descents. Edelman and Reiner and later Patricia Hersh analyzed the topology of $\Delta(P)$ in the case that $P$ is an antichain. We discuss their results and examine $\Delta(P)$ for some other classes of posets $P$.