Skip to content
DraconiticResearch
Search Sources API
181,713 studies31 sourcesUpdated 8 min ago
PreprintCC BY 4.0arXiv

Simplicial Complexes of Antichains in Root Posets and Related Combinatorics of Dyck Paths

  • Lili Mu
  • Volkmar Welker
Open at sourcePDFDOICite

Abstract

For a crystallographic root system D we consider the simplicial complex ΔD of all antichains in the root poset of D. We show that ΔD is shellable if and only if D is An, Bn, D3 or G2. Since antichains in types An and Bn can be identified with Dyck paths and symmetric Dyck paths, respectively, this yields a simplicial complex on Dyck paths. Indeed, in type An, shellability can be extended to rational Dyck paths. The f- and h-triangles then yield statistics on (symmetric/rational) Dyck paths. We determine these statistics for An and Bn and leave the case of rational Dyck paths as an open problem.

Related

  1. On the Average Sizes of Ideals and Antichains in Some Infinite Families of Posets
    Miklos Bona, Ricardo Gomez Aiza
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  2. Lyubeznik numbers of the face rings of simplicial posets
    Kosuke Shibata, Kohji Yanagawa
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Interval endomorphism algebras of posets: Reedy structure, combinatorics, and homological theory
    Toshitaka Aoki
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  4. Complementary vectors of simplicial complexes
    Matt Larson, Alan Stapledon
    Forum of Mathematics, SigmaPeer-reviewedCC BY 4.0Full text1 citedCombinatorics & discrete mathematicsCrossref
  5. Adaptive Uncertainty Quantification on Simplicial Complexes with Conformalized Simplicial Neural Networks
    Lekun Wang, Jaidev Goel, Yulia R. Gel +1
    arXivPreprintarXiv licenceFull textStatisticsarXiv

Cite

BibTeXRIS
BibTeX
@misc{mu2026simplicial,
  title = {{Simplicial Complexes of Antichains in Root Posets and Related Combinatorics of Dyck Paths}},
  author = {Lili Mu and Volkmar Welker},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14054},
  eprint = {2609.14054},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14054},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Simplicial Complexes of Antichains in Root Posets and Related Combinatorics of Dyck Paths
AU  - Lili Mu
AU  - Volkmar Welker
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.14054
UR  - https://arxiv.org/abs/2609.14054
AB  - For a crystallographic root system D we consider the simplicial complex Δ_D of all antichains in the root poset of D. We show that Δ_D is shellable if and only if D is A_n, B_n, D_3 or G_2. Since antichains in types A_n and B_n can be identified with Dyck paths and symmetric Dyck paths, respectively, this yields a simplicial complex on Dyck paths. Indeed, in type A_n, shellability can be extended to rational Dyck paths. The f- and h-triangles then yield statistics on (symmetric/rational) Dyck paths. We determine these statistics for A_n and B_n and leave the case of rational Dyck paths as an open problem.
ER  -