Strongly Signable and Partitionable Posets

The class of Strongly Signable partially ordered sets is introduced and studied. It is show that strong signability, reminiscent of Björner–Wachs’ recursive coatom orderability, provides a useful and broad sufficient condition for a poset to be dual CR and hence partitionable. The flag h-vectors of strongly signable posets are therefore non-negative. It is proved that recursively shellable posets, polyhedral fans, and face lattices of partitionable simplicial complexes are all strongly signable, and it is conjectured that all spherical posets are. It is concluded that the barycentric subdivision of a partitionable complex is again partitionable, and an algorithm for producing a partitioning of the subdivision from a partitioning of the complex is described. An expression for the flag h-polynomial of a simplicial complex in terms of its h-vector is given, and is used to demonstrate that the flag h-vector is symmetric or non-negative whenever the h-vector is.