{
  "artifactId": "artifact.library.mathlib.euler-odd-distinct-partitions.v001",
  "candidateOnly": false,
  "category": "Combinatorics",
  "claimBoundary": "This page indexes Mathlib’s cardinality form of Euler’s partition theorem. For every natural n, odds n is the finite set of partitions of n all of whose parts are odd, while distincts n is the finite set of partitions of n whose parts multiset has no repetitions. The declaration says that these two finite sets have equal cardinality. It does not itself construct an explicit bijection, enumerate either family, or state a broader partition identity.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem Nat.Partition.card_odds_eq_card_distincts (n : ℕ) : #(Nat.Partition.odds n) = #(Nat.Partition.distincts n)",
  "family": "Enumerative partition theory",
  "id": "library.mathlib.euler-odd-distinct-partitions.v001",
  "links": {
    "collection": "/collections/landmark-theorems-in-mathlib/",
    "collectionData": "/data/collections/mathlib-landmarks.json",
    "entry": "/library-theorems/euler-odd-distinct-partitions/",
    "entryData": "/data/library-theorems/euler-odd-distinct-partitions.json",
    "evidence": "/proofs/artifact.library.mathlib.euler-odd-distinct-partitions.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.euler-odd-distinct-partitions.v001.evidence.json",
    "source": "/sources/upstream/euler-odd-distinct-partitions/"
  },
  "landmark": {
    "completeFormalType": "theorem Nat.Partition.card_odds_eq_card_distincts (n : ℕ) : #(Nat.Partition.odds n) = #(Nat.Partition.distincts n)",
    "concepts": [
      "integer partitions",
      "odd parts",
      "distinct parts",
      "generating functions",
      "Glaisher’s theorem"
    ],
    "plainLanguage": "Fix a natural number n. Count the partitions of n made only from odd numbers, allowing repeats. Then count the partitions of n in which no number repeats, allowing both odd and even parts. Euler’s theorem says the two counts are always equal.",
    "poster": {
      "alt": "A portrait editorial theorem poster states the equality between counts of odd-part and distinct-part partitions and follows the checked generating-function route through Glaisher’s theorem at m equals two.",
      "byteSize": 1938706,
      "caption": "The generating functions agree, so every coefficient counts equally many odd-part and distinct-part partitions.",
      "description": "The poster defines both finite partition families, gives their cardinality equality for every natural n, and traces the Mathlib proof from restricted-partition power series through Glaisher’s product identity and coefficient comparison to the m = 2 specialization. Its scope footer states that the declaration proves a count equality rather than providing an explicit bijection.",
      "derivatives": [
        {
          "byteSize": 88802,
          "height": 960,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/euler-odd-distinct-partitions/theorem-poster-v1-640.webp",
          "sha256": "sha256:be021c01ded68c67a0c8c31ef24a57e8df0b737b1e510534140fd04babe5243c",
          "width": 640
        },
        {
          "byteSize": 161910,
          "height": 1536,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/euler-odd-distinct-partitions/theorem-poster-v1-1024.webp",
          "sha256": "sha256:2b78b25878e29283425c753dc34d834e9ee1970325a23d90c9f6705fc63c8735",
          "width": 1024
        }
      ],
      "generatedAt": "2026-07-25T16:35:50-04:00",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/euler-odd-distinct-partitions/theorem-poster-v1.png",
      "sha256": "sha256:b8119113a484d5272ddc9043d21d23e79f6320978fcff087189214c5d19d05b0",
      "title": "Euler’s Odd–Distinct Partition Theorem at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "PARTITIONS · GENERATING FUNCTIONS\nEULER’S ODD–DISTINCT PARTITION THEOREM\nFOR EVERY NATURAL NUMBER n\n#(odds n) = #(distincts n)\nODD PARTS\nEvery part is odd. Repetitions are allowed.\nDISTINCT PARTS\nNo part repeats. Even parts are allowed.\nTHE CHECKED ROUTE\n1 · Encode both families as power series\n2 · Prove Glaisher’s product identity\n3 · Compare coefficients\n4 · Specialize m = 2\nEXACT SCOPE\nEquality of cardinalities. This declaration does not supply an explicit bijection."
    },
    "visual": {
      "alt": "A text-free engraved balance holds two layered partition families: an odd-part representative with rows of lengths five, three, three, and one, and a distinct-part representative with rows of lengths six, three, two, and one.",
      "byteSize": 2361145,
      "caption": "For each natural n, the odd-part and distinct-part families have the same number of partitions.",
      "description": "A landscape editorial schematic contrasts two layered families of Ferrers-style partition cards. The exact foreground examples both total twelve: the left uses odd parts 5+3+3+1, visibly allowing repetition, while the right uses distinct parts 6+3+2+1, visibly including even parts. Blank offset silhouettes indicate the larger families, and a balanced counting instrument emphasizes equality of cardinalities without pairing individual partitions or claiming that the selected declaration supplies a bijection.",
      "derivatives": [
        {
          "byteSize": 26644,
          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/euler-odd-distinct-partitions/theorem-schematic-v1-640.webp",
          "sha256": "sha256:3bee3231bf01515469344ba6786adcc0b4daae105ef02393811bb1de0690699f",
          "width": 640
        },
        {
          "byteSize": 76986,
          "height": 800,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/euler-odd-distinct-partitions/theorem-schematic-v1-1200.webp",
          "sha256": "sha256:7c5f3f0ee3dedde572a4b996c812db8f5ab1dacb91476d3ba5e90878ef686f92",
          "width": 1200
        }
      ],
      "generatedAt": "2026-07-25T16:35:50-04:00",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/euler-odd-distinct-partitions/theorem-schematic-v1.png",
      "sha256": "sha256:acdef53cf93be061b8e44355edcfca10b279e9528337afa877fd11283685f06b",
      "title": "Two partition families balance at every total",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "Euler’s odd-versus-distinct partition identity is a foundational equinumerosity theorem in enumerative combinatorics. The Mathlib endpoint is concise, but it rests on a substantial generating-function development proving Glaisher’s more general restriction theorem and then specializing it at two."
  },
  "nonClaims": [
    "The selected declaration proves equality of cardinalities, not an explicit bijection between individual partitions.",
    "The odd-parts family allows repetitions; only the numerical value of every part must be odd.",
    "The distinct-parts family permits even parts; its restriction is that no part repeats.",
    "The checked source proves the result through generating functions and Glaisher’s theorem specialized to m = 2.",
    "ProofAtlas is indexing an existing Mathlib theorem, not claiming new mathematics."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.euler-odd-distinct-partitions.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "Current public presentation reviewed",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "For every natural n, partitions of n into odd parts and partitions of n into distinct parts are equinumerous.",
  "targetId": "target.library.mathlib.euler-odd-distinct-partitions.v001",
  "title": "Euler’s Odd–Distinct Partition Theorem",
  "upstreamOrigin": {
    "declarationName": "Nat.Partition.card_odds_eq_card_distincts",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Combinatorics/Enumerative/Partition/Glaisher.lean#L158",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
    "sourceArtifactHash": "sha256:1609afc1da7752036198d78304607aa7e1e55bbbbe6901fdb96385268b2602bb",
    "sourceByteLength": 6554,
    "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6",
    "sourceFile": "Mathlib/Combinatorics/Enumerative/Partition/Glaisher.lean",
    "sourceLine": 158,
    "verificationKind": "git_worktree"
  },
  "publicPresentationReview": {
    "status": "reviewed"
  }
}