Interpretations of the cd-index
Abstract
We extend the recent work of Caster--Guyer--Samper (2026+) to establish a combinatorial interpretation of the cd-index for all S-partitionable posets, which includes all polytopes and S-shellable spheres. This interpretation is inductive on rank, but does not require passing to the order complex of the poset. We also provide topological interpretations of each cd-monomial. Furthermore, we establish a recurrence on the cd-indices of shelling components of the d-simplex that does not require the use of André permutations, a unimodality result for cd-words ending in d for shelling components of the d-simplex and the d-cube, and provide two families of subdivisions of the boundary of the d-simplex whose sequence of cd-indices is unbounded on all but the monomial cd+1 and all but the two monomials cd+1 and dcd−1.
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BibTeX
@misc{guyer2026interpretations,
title = {{Interpretations of the $\textbfcd$-index}},
author = {Dan Guyer and Alex Waugh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25491},
eprint = {2610.25491},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25491},
}
RIS
TY - UNPB
TI - Interpretations of the cd-index
AU - Dan Guyer
AU - Alex Waugh
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25491
UR - https://arxiv.org/abs/2610.25491
AB - We extend the recent work of Caster--Guyer--Samper (2026+) to establish a combinatorial interpretation of the cd-index for all S-partitionable posets, which includes all polytopes and S-shellable spheres. This interpretation is inductive on rank, but does not require passing to the order complex of the poset. We also provide topological interpretations of each cd-monomial. Furthermore, we establish a recurrence on the cd-indices of shelling components of the d-simplex that does not require the use of André permutations, a unimodality result for cd-words ending in d for shelling components of the d-simplex and the d-cube, and provide two families of subdivisions of the boundary of the d-simplex whose sequence of cd-indices is unbounded on all but the monomial c^d+1 and all but the two monomials c^d+1 and dc^d−1.
ER -