FA-5671 / Combinatorics / Open access
Stirling second kind · case 01
Labelled assignments include empty blocks and distinguish block names.
ROOT CAUSE
Labelled assignments include empty blocks and distinguish block names.
VERIFIED 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)
Unsuccessful approach: Onto assignments still distinguish permutations of block labels.
Case 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)
Why this case matters
Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Combinatorics results depend on the stated convention.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR
N = 1
observations = []
def solve(n, k):
return k**n
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (3, 2)', solve(*(3, 2)), 3)
check('fixture 2: (4, 2)', solve(*(4, 2)), 7)
check('fixture 3: (0, 0)', solve(*(0, 0)), 1)
check('fixture 4: (2, 3)', solve(*(2, 3)), 0)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| fixture 1: (3, 2) | 8 | 3 | Failed |
| fixture 2: (4, 2) | 16 | 7 | Failed |
| fixture 3: (0, 0) | 1 | 1 | Passed |
| fixture 4: (2, 3) | 9 | 0 | Failed |
SHA-256 / 19c045e9b79d780b2ca704e348eaea7a7b65a414abd1745e768588e35a0dddbc
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR
N = 1
observations = []
def solve(n, k):
return sum((-1)**i*math.comb(k,i)*(k-i)**n for i in range(k+1))
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (3, 2)', solve(*(3, 2)), 3)
check('fixture 2: (4, 2)', solve(*(4, 2)), 7)
check('fixture 3: (0, 0)', solve(*(0, 0)), 1)
check('fixture 4: (2, 3)', solve(*(2, 3)), 0)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| fixture 1: (3, 2) | 6 | 3 | Failed |
| fixture 2: (4, 2) | 14 | 7 | Failed |
| fixture 3: (0, 0) | 1 | 1 | Passed |
| fixture 4: (2, 3) | 0 | 0 | Passed |
SHA-256 / 765163f497fb62c87b61696eac6092c76f5183693f6347a042c03255db1b87af
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR
N = 1
observations = []
def solve(n, k):
return sum((-1)**i*math.comb(k,i)*(k-i)**n for i in range(k+1))//math.factorial(k)
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (3, 2)', solve(*(3, 2)), 3)
check('fixture 2: (4, 2)', solve(*(4, 2)), 7)
check('fixture 3: (0, 0)', solve(*(0, 0)), 1)
check('fixture 4: (2, 3)', solve(*(2, 3)), 0)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| fixture 1: (3, 2) | 3 | 3 | Passed |
| fixture 2: (4, 2) | 7 | 7 | Passed |
| fixture 3: (0, 0) | 1 | 1 | Passed |
| fixture 4: (2, 3) | 0 | 0 | Passed |
SHA-256 / f93429a148b626a8ec997b3443b7b3594ae0f55624d6444f38f15192916d9afd
Verification & scope
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.
Observations recorded using Python 3.12.14 at 2026-09-29T14:37:52.216208+00:00.
Case digest / b9c0101ef9e7dfb2b13973880318b124044411945dbc8559bdf04316fb827e6d