{"abstract":"Labelled assignments include empty blocks and distinguish block names.","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. Count partitions of n labelled elements into k nonempty unlabelled blocks, with S(0,0)=1. Exact operational definition: sum((-1)**i*math.comb(k,i)*(k-i)**n for i in range(k+1))//math.factorial(k)","evaluation_group":"model-cd8ee805fb6385bc","failed_approach":"Onto assignments still distinguish permutations of block labels.","family":"xn-stirling-second-kind","id":"FA-5671","implementations":{"attempt":{"sha256":"765163f497fb62c87b61696eac6092c76f5183693f6347a042c03255db1b87af","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, k):\n    return sum((-1)**i*math.comb(k,i)*(k-i)**n for i in range(k+1))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (3, 2)', solve(*(3, 2)), 3)\ncheck('fixture 2: (4, 2)', solve(*(4, 2)), 7)\ncheck('fixture 3: (0, 0)', solve(*(0, 0)), 1)\ncheck('fixture 4: (2, 3)', solve(*(2, 3)), 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":"19c045e9b79d780b2ca704e348eaea7a7b65a414abd1745e768588e35a0dddbc","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, k):\n    return k**n\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (3, 2)', solve(*(3, 2)), 3)\ncheck('fixture 2: (4, 2)', solve(*(4, 2)), 7)\ncheck('fixture 3: (0, 0)', solve(*(0, 0)), 1)\ncheck('fixture 4: (2, 3)', solve(*(2, 3)), 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":"f93429a148b626a8ec997b3443b7b3594ae0f55624d6444f38f15192916d9afd","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, k):\n    return sum((-1)**i*math.comb(k,i)*(k-i)**n for i in range(k+1))//math.factorial(k)\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (3, 2)', solve(*(3, 2)), 3)\ncheck('fixture 2: (4, 2)', solve(*(4, 2)), 7)\ncheck('fixture 3: (0, 0)', solve(*(0, 0)), 1)\ncheck('fixture 4: (2, 3)', solve(*(2, 3)), 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-stirling-second-kind","generated_at":"2026-09-29T14:37:52.216208+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 sum((-1)**i*math.comb(k,i)*(k-i)**n for i in range(k+1))//math.factorial(k)","root_cause":"Labelled assignments include empty blocks and distinguish block names.","sha256":"b9c0101ef9e7dfb2b13973880318b124044411945dbc8559bdf04316fb827e6d","title":"Stirling second kind · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":113.499,"exit_code":1,"observations":[{"actual":6,"check":"fixture 1: (3, 2)","expected":3,"passed":false},{"actual":14,"check":"fixture 2: (4, 2)","expected":7,"passed":false},{"actual":1,"check":"fixture 3: (0, 0)","expected":1,"passed":true},{"actual":0,"check":"fixture 4: (2, 3)","expected":0,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (3, 2)\", \"actual\": 6, \"expected\": 3, \"passed\": false}, {\"check\": \"fixture 2: (4, 2)\", \"actual\": 14, \"expected\": 7, \"passed\": false}, {\"check\": \"fixture 3: (0, 0)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 4: (2, 3)\", \"actual\": 0, \"expected\": 0, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":50.576,"exit_code":1,"observations":[{"actual":8,"check":"fixture 1: (3, 2)","expected":3,"passed":false},{"actual":16,"check":"fixture 2: (4, 2)","expected":7,"passed":false},{"actual":1,"check":"fixture 3: (0, 0)","expected":1,"passed":true},{"actual":9,"check":"fixture 4: (2, 3)","expected":0,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (3, 2)\", \"actual\": 8, \"expected\": 3, \"passed\": false}, {\"check\": \"fixture 2: (4, 2)\", \"actual\": 16, \"expected\": 7, \"passed\": false}, {\"check\": \"fixture 3: (0, 0)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 4: (2, 3)\", \"actual\": 9, \"expected\": 0, \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":42.674,"exit_code":0,"observations":[{"actual":3,"check":"fixture 1: (3, 2)","expected":3,"passed":true},{"actual":7,"check":"fixture 2: (4, 2)","expected":7,"passed":true},{"actual":1,"check":"fixture 3: (0, 0)","expected":1,"passed":true},{"actual":0,"check":"fixture 4: (2, 3)","expected":0,"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (3, 2)\", \"actual\": 3, \"expected\": 3, \"passed\": true}, {\"check\": \"fixture 2: (4, 2)\", \"actual\": 7, \"expected\": 7, \"passed\": true}, {\"check\": \"fixture 3: (0, 0)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 4: (2, 3)\", \"actual\": 0, \"expected\": 0, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}