{
  "abstract": "We study diagonal billiard trajectories inside the Young diagram of an integer partition $λ$. A trajectory has slope $\\pm 1$, passes straight through sides shared by adjacent cells, and reflects from the exterior boundary until it closes. Let $σ(λ)$ be the number of closed orbits. This extends the mirror-curve model of Chokwe sona sand drawings studied by Gerdes from rectangular grids to arbitrary Young diagrams. Let $G_λ$ be the cell-adjacency graph of $λ$, with its natural planar embedding. We identify the billiard orbits with the components of the medial link of $G_λ$, and deduce that $σ(λ) = 1 + \\dim \\mathcal{B}(G_λ) = \\mathrm{nullity}\\, L(G_λ)$ over $\\mathbb{F}_2$, where $\\mathcal{B}$ is the binary bicycle space and $L$ the mod-2 Laplacian. Writing $λ^\\square$ for the diagram obtained by deleting the first row and column of $λ$, we further prove $σ(λ) = 1 + \\mathrm{nullity}_{\\mathbb{F}_2} A(G_{λ^\\square})$. For rectangles this recovers Gerdes' formula $σ(n^m) = \\gcd(m,n)$ via identities for Fibonacci polynomials over $\\mathbb{F}_2$, and in general it gives the characterization: $σ(λ) = 1$ if and only if $λ^\\square$ has an odd number of domino tilings. Using the checkerboard bipartition of $λ^\\square$, we decompose $σ(λ) - 1$ into a color-imbalance term, related to the BG-rank of Berkovich-Garvan, and an even rank-deficiency term. This yields parity restrictions and lower bounds for the orbit number, and shows that for any fixed $d$, asymptotically all partitions have more than $d$ orbits. We also prove that $σ(λ)$ is at most the Durfee length of $λ$, determine $σ(n, n-1, \\ldots, 1) = \\lceil n/2 \\rceil$ for staircase partitions, and show that the adjacency-nullity formula is independent of the ground field.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16533",
  "author_total": 1,
  "authors": [
    {
      "name": "David J. Hemmer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9srqden4qe8fn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16533",
  "fulltext_url": null,
  "id": "9srqden4qe8fn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16533",
  "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",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213413237",
  "pdf_url": "https://arxiv.org/pdf/2609.16533",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9srqden4qe8fn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16533",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16533"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Billiard Orbits in Young Diagrams: Medial Links, Bicycle Spaces, and Domino Tilings",
  "title_text": "Billiard Orbits in Young Diagrams: Medial Links, Bicycle Spaces, and Domino Tilings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/9srqden4qe8fn",
  "venue": "arXiv",
  "versions": []
}