Enumerating chains of signed order ideals

Given a finite poset P, there is an associated distributive lattice J(P) consisting of the order ideals of P ordered by inclusion. In this talk I will consider a signed analogue of this association, where the notion of an order ideal is replaced by that of a ``signed order ideal.'' The resulting poset of signed order ideals is Eulerian and EL-shellable. Its cd-index, which encodes basic chain-enumerative information, can be computed by summing entries of the flag h-vector of J(P). The proof of the latter result is based on earlier work of Billera, Ehrenborg, and Readdy establishing a similar relationship between the cd-index of an oriented matroid and the flag h-vector of the underlying matroid.