{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R706",
  "content_hash": null,
  "slug": "sbmrlc-claim-macwilliams-rank-formula",
  "type": "claim",
  "title": "MacWilliams's product formula gives every rank count",
  "summary": "The unrestricted-diagonal formula specializes cleanly to characteristic two and reproduces the candidate's enumerated rows.",
  "relevance": "For Rank log-concavity for symmetric binary matrices through order fifty, record sbmrlc-claim-macwilliams-rank-formula (“MacWilliams's product formula gives every rank count”) records a bound, answer, status fact, or structural consequence. The record states: The unrestricted-diagonal formula specializes cleanly to characteristic two and reproduces the candidate's enumerated rows.",
  "relevance_source": "recorded",
  "body": "MacWilliams's Theorem 2 counts symmetric matrices of each rank over a finite field. Lewis, Liu, Morales, Panova, Sam, and Zhang reproduce it as Equation (4.5), using \\(\\operatorname{sym}(n,r)\\) for symmetric matrices with no diagonal restriction. At \\(q=2\\), the result is\n\\[\nR_{n,r}=\n\\left(\\prod_{i=1}^{\\lfloor r/2\\rfloor}\\frac{2^{2i}}{2^{2i}-1}\\right)\n\\left(\\prod_{i=0}^{r-1}(2^{n-i}-1)\\right),\n\\]\nwith empty products equal to one. This formula supplies every exact rank vector through order 50. The executable artifact evaluates all 1,325 entries as integers, checks that each row sums to \\(2^{n(n+1)/2}\\), and records an unambiguous SHA-256 digest of the complete coefficient stream.\n\nThe diagonal convention was checked in two ways. The later paper explicitly distinguishes \\(\\operatorname{sym}(n,r)\\), with unrestricted diagonal, from \\(\\operatorname{sym}_0(n,r)\\), whose diagonal is zero. Independent enumeration of all \\(2^{n(n+1)/2}\\) upper-triangular bit assignments for every \\(1\\leq n\\leq6\\) agrees with the product formula. The first six rows are\n\\[\n\\begin{aligned}\n&(1,1),\\\\\n&(1,3,4),\\\\\n&(1,7,28,28),\\\\\n&(1,15,140,420,448),\\\\\n&(1,31,620,4340,13888,13888),\\\\\n&(1,63,2604,39060,291648,874944,888832).\n\\end{aligned}\n\\]",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "all integers n >= 0 and 0 <= r <= n for symmetric n by n matrices over F_2 with unrestricted diagonal"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1080/00029890.1969.12000160",
      "locator": "F. Jessie MacWilliams, Orthogonal Matrices Over Finite Fields, American Mathematical Monthly 76(2) (1969), 152-164, Theorem 2; Joel Brewster Lewis et al., Matrices with Restricted Entries and q-Analogues of Permutations, Journal of Combinatorics 2(3) (2011), 355-395, Equation (4.5) and the definitions preceding Proposition 4.12, arXiv:1011.4539"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1080/00029890.1969.12000160",
    "locator": "F. Jessie MacWilliams, Orthogonal Matrices Over Finite Fields, American Mathematical Monthly 76(2) (1969), 152-164, Theorem 2; Joel Brewster Lewis et al., Matrices with Restricted Entries and q-Analogues of Permutations, Journal of Combinatorics 2(3) (2011), 355-395, Equation (4.5) and the definitions preceding Proposition 4.12, arXiv:1011.4539"
  },
  "relations": [
    {
      "slug": "R707",
      "title": "The symmetric binary rank distribution is strictly log-concave",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R704",
      "title": "Replayable exact rank and log-concavity sweep",
      "object_type": "artifact",
      "relation": "uses",
      "direction": "incoming"
    },
    {
      "slug": "R705",
      "title": "The literature convention matches unrestricted binary diagonals",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "symmetric-binary-matrix-rank-log-concavity",
      "title": "symmetric binary matrix rank log concavity",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}