Sweeping the cd-Index and the Toric h-Vector

TL;DR

This paper develops formulas linking the cd-index and toric h-vector of convex polytopes through hyperplane sweepings, providing new computational methods and insights into their combinatorial structure.

Contribution

It introduces formulas for the cd-index and toric h-vector derived from hyperplane sweepings and interprets these via S-shelling of the dual polytope, also proposing an extended toric h-vector.

Findings

Derived formulas for cd-index and toric h-vector from hyperplane sweepings.

Provided a method to compute the toric h-vector directly from the cd-index.

Introduced an extended toric h-vector capturing full flag h-vector information.

Abstract

We derive formulas for the cd-index and the toric h-vector of a convex polytope P from a sweeping by a hyperplane. These arise from interpreting the corresponding S-shelling of the dual of P. We describe a partition of the faces of the complete truncation of P to reflect explicitly the nonnegativity of its cd-index and what its components are counting. One corollary is a quick way to compute the toric h-vector directly from the cd-index. We also propose an "extended toric" h-vector that fully captures the information in the flag h-vector.