{
  "abstract": "We study Motzkin paths satisfying the red-blue rule introduced by Wu, Eu, Ku and Shih. We answer their Question 8.5 and exhibit two new Baxter structures. Let $\\mathcal M_m$ denote the set of such paths of length $m$ and $a_m=|\\mathcal M_m|$. We give a combinatorial proof of \\[ a_{2n-1}=\\mathrm{A005802}(n),\\qquad a_{2n}=\\mathrm{A216947}(n) \\qquad (n\\geq 1). \\] For the odd subsequence, we construct a bijection with $1234$-avoiding permutations which sends the initial up and terminal down runs to the initial decreasing runs of a permutation and its inverse. The boundary correspondence arising from this bijection further identifies red-blue Motzkin paths of length $2n$ with $1234$-avoiding permutations of length $n+1$ in which $1$ occurs to the left of $2$. We establish parallel binomial decompositions of this restricted permutation class and of $2$-colored noncrossing set partitions. The two decompositions have the same binomial coefficients, and their respective cores are explicitly in bijection with non-intersecting triples of lattice paths, a classical Baxter family. This establishes the even identity. Moreover, we construct a second bijection between odd-length red-blue Motzkin paths and $1234$-avoiding permutations. Under this bijection, splitting a path at its ground-level blue horizontal steps corresponds to the skew decomposition of the associated permutation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06599",
  "author_total": 1,
  "authors": [
    {
      "name": "Joanna N. Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a3wbf9tkwjqm1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06599",
  "fulltext_url": null,
  "id": "a3wbf9tkwjqm1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06599",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06599",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a3wbf9tkwjqm1.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06599",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06599"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Baxter Structures and the Odd-Even Enumeration of Red-Blue Motzkin Paths",
  "title_text": "Baxter Structures and the Odd-Even Enumeration of Red-Blue Motzkin Paths",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/a3wbf9tkwjqm1",
  "venue": "arXiv",
  "versions": []
}