{
  "abstract": "How many ways are there to select $k$ objects from $n$ types when type $i$ is available in only $s_{i}$ copies? For equal bounds, this is de Moivre's problem of obtaining a prescribed total with $n$ dice, now a textbook application of the principle of inclusion-exclusion. For arbitrary bounds, the answer is encoded by the rank polynomial $\\prod_{i=1}^{n}(1+x+\\cdots+x^{s_{i}})$. Its coefficients enumerate bounded compositions and submultisets, give, after normalization, the probability mass function of a sum of independent discrete uniform random variables, and are the weight multiplicities of a tensor product of $\\mathrm{SU}(2)$ representations. Most of these interpretations are classical but dispersed; here we place them in a common notation and document their provenance. A complement involution proves symmetry and boundary identities. For arbitrary bounds, lower-half first differences give both symmetric-chain starts and mixed-spin multiplicities, and the chain-length and irreducible-dimension profiles agree. Convolution gives log-concavity and the central mode. For equal bounds with fixed $s \\geq 1$ and even $ns$, local expansions give central asymptotics with explicit $1/n$ and $1/n^{2}$ corrections for the coefficient and its first difference. A Poisson limit gives the first boundary crossover of these differences at scale $k \\sim λn^{s/(s+1)}$, with factor $e^{-αλ^{s+1}}$ when the proportion of bounds equal to the fixed minimum $s \\geq 1$ tends to $α>0$. Central first differences include the $s$-Catalan numbers, themselves Littlewood-Richardson coefficients; central layers yield maximum-size constant-sum codebooks for hash-based signatures. Under the polynomial binomial convention, the full inclusion-exclusion transform vanishes, and the omitted terms sum to the negative of the number of bounded compositions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31865",
  "author_total": 2,
  "authors": [
    {
      "name": "Hacène Belbachir"
    },
    {
      "name": "José Ricardo G. Mendonça"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/70ymwbdwasm1b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31865",
  "fulltext_url": null,
  "id": "70ymwbdwasm1b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31865",
  "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-ph"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223694615",
  "pdf_url": "https://arxiv.org/pdf/2610.31865",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/70ymwbdwasm1b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:15Z",
      "native_id": "2610.31865",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31865"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Bounded compositions and products of chains, or bi(s)nomials everywhere from de Moivre's formula to spin multiplicities and constant-sum codes",
  "title_text": "Bounded compositions and products of chains, or bi(s)nomials everywhere from de Moivre's formula to spin multiplicities and constant-sum codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/70ymwbdwasm1b",
  "venue": "arXiv",
  "versions": []
}