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Poset Partitions and the Combinatorics of the $\textbf{cd}$-Index

Felipe Caster, Dan Guyer, José Alejandro Samper

TL;DR

The paper develops the notions of S-partitionable (Eulerian) and SE-partitionable (semi-Eulerian) posets to provide combinatorial proofs of non-negativity for the cd-index. It extends Stanley’s S-shellability by introducing recursive partition frameworks that decompose the cd-index into non-negative contributions from smaller rank components, via pre-blocks and coatom-wise partitions. The SE-extension adapts the framework to handle the Euler characteristic corrections needed for semi-Eulerian posets, yielding analogous non-negativity results and enabling a wide range of non-Eulerian examples to be treated. Together, these results offer explicit recursive formulas for the cd-index, illuminate structural consequences, and suggest broad future directions for understanding cd-positivity in more general poset and manifold contexts.

Abstract

We introduce a new class of Eulerian posets, called S-partitionable posets, which have a non-negative cd-index. These posets are a generalization of S-shellable complexes introduced by Stanley in 1994. We prove that S-partitionable posets have a non-negative cd-index via a recursive formula. Then, we introduce a semi-Eulerian version of S-partitionable posets, which we call SE-partitionable posets. We show that SE-partitionable posets also have a non-negative semi-Eulerian cd-index as defined by Juhnke-Kubitzke, Samper and Venturello in 2024.

Poset Partitions and the Combinatorics of the $\textbf{cd}$-Index

TL;DR

The paper develops the notions of S-partitionable (Eulerian) and SE-partitionable (semi-Eulerian) posets to provide combinatorial proofs of non-negativity for the cd-index. It extends Stanley’s S-shellability by introducing recursive partition frameworks that decompose the cd-index into non-negative contributions from smaller rank components, via pre-blocks and coatom-wise partitions. The SE-extension adapts the framework to handle the Euler characteristic corrections needed for semi-Eulerian posets, yielding analogous non-negativity results and enabling a wide range of non-Eulerian examples to be treated. Together, these results offer explicit recursive formulas for the cd-index, illuminate structural consequences, and suggest broad future directions for understanding cd-positivity in more general poset and manifold contexts.

Abstract

We introduce a new class of Eulerian posets, called S-partitionable posets, which have a non-negative cd-index. These posets are a generalization of S-shellable complexes introduced by Stanley in 1994. We prove that S-partitionable posets have a non-negative cd-index via a recursive formula. Then, we introduce a semi-Eulerian version of S-partitionable posets, which we call SE-partitionable posets. We show that SE-partitionable posets also have a non-negative semi-Eulerian cd-index as defined by Juhnke-Kubitzke, Samper and Venturello in 2024.
Paper Structure (14 sections, 11 theorems, 30 equations, 14 figures)

This paper contains 14 sections, 11 theorems, 30 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. S-partitionable Posets
  4. A Motivating Example
  5. S-partitionable Definition
  6. S-partitions and the cd-index
  7. Consequences
  8. Semi-Eulerian Partitionability
  9. A Motivating Example
  10. SE-Partitionable posets
  11. SE-partitionable non-negativity
  12. Consequences
  13. Future Directions
  14. Acknowledgements

Key Result

Theorem 2.10

A graded poset has a cd-index if and only if its flag $f$-vector satisfies the generalized Dehn--Sommerville equations. In particular, every Eulerian poset has a cd-index.

Figures (14)

  • Figure 1: Polytope $Q$.
  • Figure 2: Shelling of $Q$.
  • Figure 3: S-partition of $Q$.
  • Figure 4: Contribution of the S-partition classes of $Q$.
  • Figure 5: A partition class and its pre-blocks,
  • ...and 9 more figures

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Theorem 2.10: Bayer1991
  • ...and 32 more