{
  "schemaVersion": "1.0",
  "discourse": {
    "id": "combinatorics",
    "name": "Combinatorics",
    "subject": "discrete_mathematics",
    "variant": "classical",
    "description": "Counting principles: sum and product rules, permutations (with/without repetition), combinations, binomial theorem, pigeonhole principle, inclusion-exclusion, derangements.",
    "structure": {
      "axioms": 0,
      "definitions": 3,
      "theorems": 11
    }
  },
  "metadata": {
    "created": "2026-03-15",
    "lastUpdated": "2026-03-15",
    "version": "1.0.0",
    "license": "CC BY 4.0",
    "authors": [
      "Welz, G."
    ],
    "methodology": "Programming Framework",
    "citation": "Welz, G. (2026). Combinatorics Dependency Graph. Programming Framework.",
    "keywords": [
      "combinatorics",
      "permutations",
      "combinations",
      "counting",
      "binomial theorem"
    ]
  },
  "sources": [
    {
      "id": "dmoi",
      "type": "primary",
      "title": "Discrete Mathematics: An Open Introduction",
      "url": "https://discrete.openmathbooks.org/dmoi4/sec_counting-combperm.html",
      "notes": "Counting principles"
    },
    {
      "id": "mathisfun",
      "type": "digital",
      "title": "Combinations and Permutations",
      "url": "https://www.mathsisfun.com/combinatorics/combinations-permutations.html",
      "notes": "Formulas"
    }
  ],
  "nodes": [
    {
      "id": "DefFact",
      "type": "definition",
      "label": "Factorial: n! = n(n-1)...1, 0!=1",
      "shortLabel": "DefFact",
      "short": "Factorial",
      "colorClass": "definition"
    },
    {
      "id": "DefSum",
      "type": "definition",
      "label": "Sum principle: disjoint choices add (OR)",
      "shortLabel": "DefSum",
      "short": "Sum principle",
      "colorClass": "definition"
    },
    {
      "id": "DefProd",
      "type": "definition",
      "label": "Product principle: sequential choices multiply (AND)",
      "shortLabel": "DefProd",
      "short": "Product principle",
      "colorClass": "definition"
    },
    {
      "id": "PermNoRep",
      "type": "theorem",
      "label": "P(n,r) = n!/(n-r)! arrangements of r from n",
      "shortLabel": "PermNoRep",
      "short": "Permutations no rep",
      "colorClass": "theorem"
    },
    {
      "id": "PermRep",
      "type": "theorem",
      "label": "n^r arrangements of r from n with repetition",
      "shortLabel": "PermRep",
      "short": "Permutations with rep",
      "colorClass": "theorem"
    },
    {
      "id": "CombNoRep",
      "type": "theorem",
      "label": "C(n,r) = n!/(r!(n-r)!) = P(n,r)/r!",
      "shortLabel": "CombNoRep",
      "short": "Combinations",
      "colorClass": "theorem"
    },
    {
      "id": "CombRep",
      "type": "theorem",
      "label": "C(n+r-1,r) ways to choose r from n with rep",
      "shortLabel": "CombRep",
      "short": "Combinations with rep",
      "colorClass": "theorem"
    },
    {
      "id": "BinomThm",
      "type": "theorem",
      "label": "(a+b)^n = sum C(n,k) a^k b^(n-k)",
      "shortLabel": "BinomThm",
      "short": "Binomial theorem",
      "colorClass": "theorem"
    },
    {
      "id": "Pascal",
      "type": "theorem",
      "label": "C(n,k) = C(n-1,k-1) + C(n-1,k)",
      "shortLabel": "Pascal",
      "short": "Pascal identity",
      "colorClass": "theorem"
    },
    {
      "id": "Pigeonhole",
      "type": "theorem",
      "label": "n+1 objects in n boxes implies one box has 2+",
      "shortLabel": "Pigeonhole",
      "short": "Pigeonhole principle",
      "colorClass": "theorem"
    },
    {
      "id": "InclExcl",
      "type": "theorem",
      "label": "|A union B| = |A| + |B| - |A intersect B|",
      "shortLabel": "InclExcl",
      "short": "Inclusion-exclusion",
      "colorClass": "theorem"
    },
    {
      "id": "InclExcl3",
      "type": "theorem",
      "label": "Inclusion-exclusion for 3 sets",
      "shortLabel": "InclExcl3",
      "short": "Incl-excl 3 sets",
      "colorClass": "theorem"
    },
    {
      "id": "Derange",
      "type": "theorem",
      "label": "D(n) = n! sum (-1)^k/k! derangements",
      "shortLabel": "Derange",
      "short": "Derangements",
      "colorClass": "theorem"
    },
    {
      "id": "Stirling2",
      "type": "theorem",
      "label": "S(n,k) = partitions of n into k nonempty sets",
      "shortLabel": "Stirling2",
      "short": "Stirling numbers",
      "colorClass": "theorem"
    }
  ],
  "edges": [
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    {
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    {
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    {
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    {
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  ],
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