The Rank-Generating Functions of Upho Posets

TL;DR

This paper explores the properties of upho posets, constructing examples with specific algebraic features, categorizing planar cases, and demonstrating the existence of an uncomputable rank-generating function, advancing understanding of this poset class.

Contribution

It introduces new upho posets with Schur-positive Ehrenborg functions, classifies planar upho posets' rank-generating functions, and proves the existence of an uncomputable case.

Findings

Constructed upho posets with Schur-positive Ehrenborg functions

Categorized rank-generating functions of planar upho posets

Proved existence of an upho poset with uncomputable rank-generating function

Abstract

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.