ETD

• cd-index
• Chow functions
• Eulerian posets
• KLS theory of posets
• polytopes
• stellahedral transform

Eulerian posets form a special class of partially ordered sets which are often encountered in combinatorics and provide a natural setting for generalizing many results in polyhedral combinatorics. This thesis investigates the theory of Eulerian posets and its relation to polytopes, with particular attention to the framework of the Kazhdan–Lusztig–Stanley (KLS) theory of posets.
In chapter 1, we introduce the fundamental notions of poset and polytope theory. In chapter 2, we define Eulerian posets and develop their basic properties. We explore the geometric and topological settings in which they arise, discuss various generalizations of the Dehn-Sommerville equations, and then construct the cd-index. In chapter 3, we explain how to generalize the construction of the h-vector of simplicial polytopes into the toric h-vector of Eulerian posets. We then describe how to compute the toric h-vector from the cd-index.
In chapter 4, we introduce the KLS theory of posets and some of its invariants: the Z-function, the Chow function and the augmented Chow functions. We then study these invariants in the Eulerian case. We present results from joint work with Luis Ferroni and Tommaso Faustini in which we provide a combinatorial and geometrical interpretation of Eulerian augmented Chow polynomials. Finally, we express the previous invariants in terms of the cd-index.