Finite Eulerian posets which are binomial, Sheffer or triangular

TL;DR

This paper classifies and characterizes finite Eulerian posets that are binomial, Sheffer, or triangular, linking their structure to generating functions and geometry, and answering longstanding questions in the field.

Contribution

It provides a complete classification of Eulerian binomial posets, an almost complete classification of Eulerian Sheffer posets, and studies Eulerian triangular posets, advancing the understanding of their structure.

Findings

Complete classification of Eulerian binomial posets

Almost complete classification of Eulerian Sheffer posets

Structural determination of Eulerian triangular posets

Abstract

In this paper we study finite Eulerian posets which are binomial, Sheffer or triangular. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows: We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets; We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases; In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets. We also study Eulerian triangular posets. This paper answers questions asked by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the boolean lattice by looking at smaller intervals.