Generalized shellings for balanced simplicial complexes|
Abstract: Inspired by results on graded planar partially ordered sets (jointly obtained with Louis Billera), we introduce for each directed graph G on n vertices a generalized notion of shellability of balanced (n-1)-dimensional simplicial complexes. For the empty graph we recover the usual notion of shellability, while fixing an undirected path makes the left-to-right enumeration of the maximal chains in a planar graded partially ordered set a G-shelling. In all cases, the closed cone generated by the flag f-vectors of all G-shellable complexes turns out to be an orthant, and we obtain similar descriptions for certain intersections of G-shellability classes. Our results depend in part on the fact that every interval of a partial order induced by leaks along the edges of a graph is an upper semidistributive lattice. The Moebius inversion formula for these intervals, together with further graph-theoretic observations, yield ``graphical generalizations'' of the sieve formula. The theory seems to be generalizable to a situation when the role of the graph is taken over by a function on pairs of subsets of the vertices, satisfying a few axioms.