{
  "abstract": "Let $E$ be the exterior algebra of an $n$-dimensional vector space over a field, let $F$ be a graded free $E$-module with $r$ generators of degrees $d_1 \\le \\dots \\le d_r$, and let $M$ range over the monomial submodules of Amata and Crupi. The Hilbert function of $F/M$ is a shifted sum of $f$-vectors of simplicial complexes; the companion paper determines the convex hull of such sums, and here we count them. For $r = 1$ the count is Linusson's (1999); we show that no new recursion is needed for general $r$. Using two statistics of an $f$-vector -- its top nonvanishing degree and the initial degree of the corresponding ideal -- we prove that the number of achievable Hilbert functions is a product of $r-1$ transfer matrices, one per generator-degree gap, with entries differences of Linusson's refined counts $E^p(n,k)$. The rank enters only as the length of a matrix product. The same holds for the proper variant. Consequences: a linear recurrence in $r$ for constant gaps, with exact exponential asymptotics $K λ^r$ when the gap is positive; polynomiality in $r$ when all gaps vanish; a threshold beyond which the count is a plain power; and polynomiality in the number of vertices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14985",
  "author_total": 1,
  "authors": [
    {
      "name": "Jan Snellman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7hbc0fh7whwm3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14985",
  "fulltext_url": null,
  "id": "7hbc0fh7whwm3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14985",
  "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"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7221787717",
  "pdf_url": "https://arxiv.org/pdf/2610.14985",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7hbc0fh7whwm3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14985",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14985"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hilbert functions over the exterior algebra and f-vectors in higher rank II: Generalizing Linusson's count to sumsets of f-vectors",
  "title_text": "Hilbert functions over the exterior algebra and f-vectors in higher rank II: Generalizing Linusson's count to sumsets of f-vectors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/7hbc0fh7whwm3",
  "venue": "arXiv",
  "versions": []
}