Maximal chain descent orders

This paper introduces a partial order on the maximal chains of any finite bounded poset $P$ which has a CL-labeling $\lambda$. We call this the maximal chain descent order induced by $\lambda$, denoted $P_{\lambda}(2)$. As a first example, letting $P$ be the Boolean lattice and $\lambda$ its standard EL-labeling gives $P_{\lambda}(2)$ isomorphic to the weak order of type A. We discuss in depth other seemingly well-structured examples: the max-min EL-labeling of the partition lattice gives maximal chain descent order isomorphic to a partial order on certain labeled trees, and particular cases of the linear extension EL-labelings of finite distributive lattices produce maximal chain descent orders isomorphic to partial orders on standard Young tableaux. We observe that the order relations which one might expect to be the cover relations, those given by the "polygon moves" whose transitive closure defines the maximal chain descent order, are not always cover relations. Several examples illustrate this fact. Nonetheless, we characterize the EL-labelings for which every polygon move gives a cover relation, and we prove many well known EL-labelings do have the expected cover relations. One motivation for $P_{\lambda}(2)$ is that its linear extensions give all of the shellings of the order complex of $P$ whose restriction maps are defined by the descents with respect to $\lambda$. This yields strictly more shellings of $P$ than the lexicographic ones induced by $\lambda$. Thus, the maximal chain descent order $P_{\lambda}(2)$ might be thought of as encoding the structure of the set of shellings induced by $\lambda$.