  Upper homogeneous finite type (upho) posets are a large class of partially
ordered sets with the property that the principal order filter at every vertex
is isomorphic to the whole poset. Well-known examples include k-array trees,
the grid graphs, and the Stern poset. Very little is known about upho posets in
general. In this paper, we construct upho posets with Schur-positive Ehrenborg
quasisymmetric functions, whose rank-generating functions have rational poles
and zeros. We also categorize the rank-generating functions of all planar upho
posets. Finally, we prove the existence of an upho poset with uncomputable
rank-generating function.
