Level Eulerian Posets

TL;DR

This paper introduces level Eulerian posets, characterizes their properties, and develops algebraic and combinatorial tools to analyze their structure, including conditions for Eulerianness and shellability, and rationality of their series.

Contribution

It defines level Eulerian posets, determines Eulerianness verification bounds, and develops algebraic methods for analyzing their series and shellability.

Findings

Longest interval for Eulerianness verification is bounded for indecomposable matrices

All level Eulerian posets with indecomposable matrices have even order

The ab-series and cd-series are rational generating functions

Abstract

The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite graph occurs. When the adjacency matrix is indecomposable, we determine the length of the longest interval one needs to check to verify Eulerianness. Furthermore, we show that every level Eulerian poset associated to an indecomposable matrix has even order. A condition for verifying shellability is introduced and is automated using the algebra of walks. Applying the Skolem--Mahler--Lech theorem, the ${\bf ab}$-series of a level poset is shown to be a rational generating function in the non-commutative variables ${\bf a}$ and ${\bf b}$. In the case the poset is also Eulerian, the analogous result holds for the ${\bf cd}$-series. Using coalgebraic techniques a method is developed to recognize the ${\bf cd}$-series matrix of a level Eulerian poset.