The rank-generating functions of upho posets

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