garywelz/programming_framework

1{2  "schemaVersion": "1.0",3  "discourse": {4    "id": "combinatorics",5    "name": "Combinatorics",6    "subject": "discrete_mathematics",7    "variant": "classical",8    "description": "Counting principles: sum and product rules, permutations (with/without repetition), combinations, binomial theorem, pigeonhole principle, inclusion-exclusion, derangements.",9    "structure": {10      "axioms": 0,11      "definitions": 3,12      "theorems": 1113    }14  },15  "metadata": {16    "created": "2026-03-15",17    "lastUpdated": "2026-03-15",18    "version": "1.0.0",19    "license": "CC BY 4.0",20    "authors": [21      "Welz, G."22    ],23    "methodology": "Programming Framework",24    "citation": "Welz, G. (2026). Combinatorics Dependency Graph. Programming Framework.",25    "keywords": [26      "combinatorics",27      "permutations",28      "combinations",29      "counting",30      "binomial theorem"31    ]32  },33  "sources": [34    {35      "id": "dmoi",36      "type": "primary",37      "title": "Discrete Mathematics: An Open Introduction",38      "url": "https://discrete.openmathbooks.org/dmoi4/sec_counting-combperm.html",39      "notes": "Counting principles"40    },41    {42      "id": "mathisfun",43      "type": "digital",44      "title": "Combinations and Permutations",45      "url": "https://www.mathsisfun.com/combinatorics/combinations-permutations.html",46      "notes": "Formulas"47    }48  ],49  "nodes": [50    {51      "id": "DefFact",52      "type": "definition",53      "label": "Factorial: n! = n(n-1)...1, 0!=1",54      "shortLabel": "DefFact",55      "short": "Factorial",56      "colorClass": "definition"57    },58    {59      "id": "DefSum",60      "type": "definition",61      "label": "Sum principle: disjoint choices add (OR)",62      "shortLabel": "DefSum",63      "short": "Sum principle",64      "colorClass": "definition"65    },66    {67      "id": "DefProd",68      "type": "definition",69      "label": "Product principle: sequential choices multiply (AND)",70      "shortLabel": "DefProd",71      "short": "Product principle",72      "colorClass": "definition"73    },74    {75      "id": "PermNoRep",76      "type": "theorem",77      "label": "P(n,r) = n!/(n-r)! arrangements of r from n",78      "shortLabel": "PermNoRep",79      "short": "Permutations no rep",80      "colorClass": "theorem"81    },82    {83      "id": "PermRep",84      "type": "theorem",85      "label": "n^r arrangements of r from n with repetition",86      "shortLabel": "PermRep",87      "short": "Permutations with rep",88      "colorClass": "theorem"89    },90    {91      "id": "CombNoRep",92      "type": "theorem",93      "label": "C(n,r) = n!/(r!(n-r)!) = P(n,r)/r!",94      "shortLabel": "CombNoRep",95      "short": "Combinations",96      "colorClass": "theorem"97    },98    {99      "id": "CombRep",100      "type": "theorem",101      "label": "C(n+r-1,r) ways to choose r from n with rep",102      "shortLabel": "CombRep",103      "short": "Combinations with rep",104      "colorClass": "theorem"105    },106    {107      "id": "BinomThm",108      "type": "theorem",109      "label": "(a+b)^n = sum C(n,k) a^k b^(n-k)",110      "shortLabel": "BinomThm",111      "short": "Binomial theorem",112      "colorClass": "theorem"113    },114    {115      "id": "Pascal",116      "type": "theorem",117      "label": "C(n,k) = C(n-1,k-1) + C(n-1,k)",118      "shortLabel": "Pascal",119      "short": "Pascal identity",120      "colorClass": "theorem"121    },122    {123      "id": "Pigeonhole",124      "type": "theorem",125      "label": "n+1 objects in n boxes implies one box has 2+",126      "shortLabel": "Pigeonhole",127      "short": "Pigeonhole principle",128      "colorClass": "theorem"129    },130    {131      "id": "InclExcl",132      "type": "theorem",133      "label": "|A union B| = |A| + |B| - |A intersect B|",134      "shortLabel": "InclExcl",135      "short": "Inclusion-exclusion",136      "colorClass": "theorem"137    },138    {139      "id": "InclExcl3",140      "type": "theorem",141      "label": "Inclusion-exclusion for 3 sets",142      "shortLabel": "InclExcl3",143      "short": "Incl-excl 3 sets",144      "colorClass": "theorem"145    },146    {147      "id": "Derange",148      "type": "theorem",149      "label": "D(n) = n! sum (-1)^k/k! derangements",150      "shortLabel": "Derange",151      "short": "Derangements",152      "colorClass": "theorem"153    },154    {155      "id": "Stirling2",156      "type": "theorem",157      "label": "S(n,k) = partitions of n into k nonempty sets",158      "shortLabel": "Stirling2",159      "short": "Stirling numbers",160      "colorClass": "theorem"161    }162  ],163  "edges": [164    {165      "from": "DefFact",166      "to": "PermNoRep"167    },168    {169      "from": "DefProd",170      "to": "PermNoRep"171    },172    {173      "from": "DefProd",174      "to": "PermRep"175    },176    {177      "from": "PermNoRep",178      "to": "CombNoRep"179    },180    {181      "from": "DefFact",182      "to": "CombNoRep"183    },184    {185      "from": "CombNoRep",186      "to": "CombRep"187    },188    {189      "from": "CombNoRep",190      "to": "BinomThm"191    },192    {193      "from": "CombNoRep",194      "to": "Pascal"195    },196    {197      "from": "DefSum",198      "to": "Pigeonhole"199    },200    {201      "from": "DefSum",202      "to": "InclExcl"203    },204    {205      "from": "InclExcl",206      "to": "InclExcl3"207    },208    {209      "from": "InclExcl",210      "to": "Derange"211    },212    {213      "from": "PermNoRep",214      "to": "Derange"215    },216    {217      "from": "DefSum",218      "to": "Stirling2"219    },220    {221      "from": "DefProd",222      "to": "Stirling2"223    }224  ],225  "colorScheme": {226    "axiom": {227      "fill": "#e74c3c",228      "stroke": "#c0392b"229    },230    "definition": {231      "fill": "#3498db",232      "stroke": "#2980b9"233    },234    "theorem": {235      "fill": "#1abc9c",236      "stroke": "#16a085"237    }238  }239}