{
  "abstract": "For a crystallographic root system ${\\mathfrak D}$ we consider the simplicial complex $Δ_{\\mathfrak D}$ of all antichains in the root poset of ${\\mathfrak D}$. We show that $Δ_{\\mathfrak D}$ is shellable if and only if ${\\mathfrak D}$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14054",
  "author_total": 2,
  "authors": [
    {
      "name": "Lili Mu"
    },
    {
      "name": "Volkmar Welker"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/esbdrm709m0xc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14054",
  "fulltext_url": null,
  "id": "esbdrm709m0xc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14054",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.14054",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/esbdrm709m0xc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14054",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14054"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Simplicial Complexes of Antichains in Root Posets and Related Combinatorics of Dyck Paths",
  "title_text": "Simplicial Complexes of Antichains in Root Posets and Related Combinatorics of Dyck Paths",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/esbdrm709m0xc",
  "venue": "arXiv",
  "versions": []
}