{"abstract":"Zero components are admitted in a positive composition.","category":"Combinatorics","checks":4,"contract":"Nonnegative integer counts. Permutation and combination counts are zero when a requested selection exceeds the available objects. Shape sizes satisfy the preconditions stated below. parts>=1; count ordered vectors of parts strictly positive integers summing to n. Exact operational definition: math.comb(n-1,parts-1) if n>=parts else 0","evaluation_group":"model-0c0be955855f5985","failed_approach":"Choosing item positions does not choose gaps between items.","family":"xn-positive-composition-count","id":"FA-5681","implementations":{"attempt":{"sha256":"e1563f42175afe3501525f967bd40cc553ebbbea312d2c48d3c8825d5dbf9fd8","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport calendar\nimport statistics\nimport itertools\nfrom fractions import Fraction\nfrom datetime import date, datetime, timedelta, timezone\nfrom decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR\n\nN = 1\nobservations = []\ndef solve(n, parts):\n    return math.comb(n,parts)\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (5, 2)', solve(*(5, 2)), 4)\ncheck('fixture 2: (3, 3)', solve(*(3, 3)), 1)\ncheck('fixture 3: (2, 3)', solve(*(2, 3)), 0)\ncheck('fixture 4: (0, 1)', solve(*(0, 1)), 0)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"adfe57ecd002d9e2691862cb6b9954d0cb732f7b7cd6eb542432df4ee4278fb3","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport calendar\nimport statistics\nimport itertools\nfrom fractions import Fraction\nfrom datetime import date, datetime, timedelta, timezone\nfrom decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR\n\nN = 1\nobservations = []\ndef solve(n, parts):\n    return math.comb(n+parts-1,parts-1)\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (5, 2)', solve(*(5, 2)), 4)\ncheck('fixture 2: (3, 3)', solve(*(3, 3)), 1)\ncheck('fixture 3: (2, 3)', solve(*(2, 3)), 0)\ncheck('fixture 4: (0, 1)', solve(*(0, 1)), 0)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"fixed":{"sha256":"d66e0ae1d0f0b20abfd783d8ee09434f30a5b901b2104f94c26c79e9aea0b73f","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport calendar\nimport statistics\nimport itertools\nfrom fractions import Fraction\nfrom datetime import date, datetime, timedelta, timezone\nfrom decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR\n\nN = 1\nobservations = []\ndef solve(n, parts):\n    return math.comb(n-1,parts-1) if n>=parts else 0\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (5, 2)', solve(*(5, 2)), 4)\ncheck('fixture 2: (3, 3)', solve(*(3, 3)), 1)\ncheck('fixture 3: (2, 3)', solve(*(2, 3)), 0)\ncheck('fixture 4: (0, 1)', solve(*(0, 1)), 0)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":" This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"xn-positive-composition-count","generated_at":"2026-09-29T14:37:52.552292+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Combinatorics results depend on the stated convention.","repair":"Apply the specified mathematical contract directly, preserving all terms and boundary cases: return math.comb(n-1,parts-1) if n>=parts else 0","root_cause":"Zero components are admitted in a positive composition.","sha256":"b4d7167fa1a356913fe362eeec4ee8b24c9a966b7d5bc799b49373e09ab6d79b","title":"Positive composition count · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":47.803,"exit_code":1,"observations":[{"actual":10,"check":"fixture 1: (5, 2)","expected":4,"passed":false},{"actual":1,"check":"fixture 2: (3, 3)","expected":1,"passed":true},{"actual":0,"check":"fixture 3: (2, 3)","expected":0,"passed":true},{"actual":0,"check":"fixture 4: (0, 1)","expected":0,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (5, 2)\", \"actual\": 10, \"expected\": 4, \"passed\": false}, {\"check\": \"fixture 2: (3, 3)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 3: (2, 3)\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"fixture 4: (0, 1)\", \"actual\": 0, \"expected\": 0, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":46.336,"exit_code":1,"observations":[{"actual":6,"check":"fixture 1: (5, 2)","expected":4,"passed":false},{"actual":10,"check":"fixture 2: (3, 3)","expected":1,"passed":false},{"actual":6,"check":"fixture 3: (2, 3)","expected":0,"passed":false},{"actual":1,"check":"fixture 4: (0, 1)","expected":0,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (5, 2)\", \"actual\": 6, \"expected\": 4, \"passed\": false}, {\"check\": \"fixture 2: (3, 3)\", \"actual\": 10, \"expected\": 1, \"passed\": false}, {\"check\": \"fixture 3: (2, 3)\", \"actual\": 6, \"expected\": 0, \"passed\": false}, {\"check\": \"fixture 4: (0, 1)\", \"actual\": 1, \"expected\": 0, \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":155.219,"exit_code":0,"observations":[{"actual":4,"check":"fixture 1: (5, 2)","expected":4,"passed":true},{"actual":1,"check":"fixture 2: (3, 3)","expected":1,"passed":true},{"actual":0,"check":"fixture 3: (2, 3)","expected":0,"passed":true},{"actual":0,"check":"fixture 4: (0, 1)","expected":0,"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (5, 2)\", \"actual\": 4, \"expected\": 4, \"passed\": true}, {\"check\": \"fixture 2: (3, 3)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 3: (2, 3)\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"fixture 4: (0, 1)\", \"actual\": 0, \"expected\": 0, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}