University of Minnesota Combinatorics Seminar: 2022

Date: 04/29/2022

Speaker: Sam Hopkins

Rank and Characteristic Generating Functions of Upper Homogeneous Posets
The rank and characteristic generating functions of finite and finite type graded posets often encode combinatorially significant sequences, such as the Stirling numbers. In this talk I will discuss the rank and characteristic generating functions of upper homogeneous posets, a class of infinite posets recently introduced by Stanley. A poset is upper homogeneous (or "upho") if every principal order filter is isomorphic to the whole poset. As I will explain, one can easily show that the rank and characteristic generating functions of an upho poset are multiplicative inverses of one another. Consequently, the rank generating function of an upho lattice is the inverse of a polynomial: namely, the characteristic polynomial of the finite sublattice below the join of all atoms. It is an interesting problem to try to "go in the other direction," i.e., start with a finite graded lattice and extend it to an upho lattice. I will end by reviewing some examples and posing some questions about this problem.

Rank and Characteristic Generating Functions of Upper Homogeneous Posets

The rank and characteristic generating functions of finite and finite type graded posets often encode combinatorially significant sequences, such as the Stirling numbers. In this talk I will discuss the rank and characteristic generating functions of upper homogeneous posets, a class of infinite posets recently introduced by Stanley. A poset is upper homogeneous (or "upho") if every principal order filter is isomorphic to the whole poset. As I will explain, one can easily show that the rank and characteristic generating functions of an upho poset are multiplicative inverses of one another. Consequently, the rank generating function of an upho lattice is the inverse of a polynomial: namely, the characteristic polynomial of the finite sublattice below the join of all atoms. It is an interesting problem to try to "go in the other direction," i.e., start with a finite graded lattice and extend it to an upho lattice. I will end by reviewing some examples and posing some questions about this problem.