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

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.

💡 Research Summary

The paper introduces a new class of Eulerian posets called S‑partitionable posets and proves that their cd‑index is always non‑negative. The authors begin by reviewing the cd‑index, a compact encoding of the flag f‑vector of a graded poset, and recall that while many Eulerian posets have non‑negative cd‑coefficients, there exist Eulerian examples with negative coefficients. Stanley’s 1994 notion of S‑shellability gave a combinatorial proof of non‑negativity for a large family (including all polytopal spheres), but the shellability argument does not provide a direct combinatorial interpretation of the cd‑coefficients.

Inspired by the relationship between shellability and partitionability for simplicial complexes, the authors define S‑partitionability as a purely combinatorial partition of the elements of a bounded Eulerian poset (excluding the top element) into classes indexed by the coatoms, each class satisfying a recursive property that mirrors the inductive step in Stanley’s shellability proof. This partition extracts the essential features needed to guarantee that each coatom contributes a non‑negative sum of cd‑words to the overall cd‑index. The main result, Theorem A, shows that the cd‑index of an S‑partitionable poset can be computed recursively from the cd‑indices of its lower‑rank S‑partitionable pieces, and consequently all its coefficients are non‑negative. The proof adapts Lee’s decomposition of the cd‑index for polytopes, replacing geometric arguments with combinatorial analogues based on the partition structure.

The paper then extends the framework to semi‑Eulerian posets, which arise when the global Euler characteristic matches that of a sphere but local intervals may fail the Eulerian condition (e.g., even‑dimensional manifolds). Juhnke‑Kubitzke, Samp­er, and Venturello (2024) defined a modified flag f‑vector and a corresponding cd‑index for such posets. To handle this setting, the authors introduce SE‑partitionability, a slight relaxation of S‑partitionability that incorporates the necessary correction terms in the recursive step. Every S‑partitionable poset is automatically SE‑partitionable, but the new notion also captures many non‑Eulerian examples, such as triangulations of two‑dimensional semi‑Eulerian pseudomanifolds and any simplicial semi‑Eulerian complex that admits a partition. Theorem B establishes that SE‑partitionable posets have a recursively computable, non‑negative semi‑Eulerian cd‑index.

Throughout, the authors provide concrete examples illustrating that S‑partitionability imposes far weaker topological constraints than shellability: while S‑shellable posets have order complexes homeomorphic to spheres, S‑partitionable posets can be homology manifolds of various types. They also discuss how their combinatorial tools give heuristic reasons to expect positivity (or at least useful lower bounds) for the cd‑index of regular CW‑homology manifolds, a class previously out of reach for algebraic methods such as those used by Karu.

The paper concludes with several open problems: developing algorithmic criteria for detecting S‑ or SE‑partitionability, exploring inequalities among cd‑coefficients beyond non‑negativity, and investigating connections between the correction terms in the semi‑Eulerian case and other topological invariants. By providing a purely combinatorial pathway to cd‑index positivity for both Eulerian and semi‑Eulerian posets, the work broadens the toolkit available for studying flag enumerative invariants in poset topology and suggests promising directions for future research.