FA-15221 / Numerics / Open access

Multiset permutation unrank: duplicate block normalization · case 01

The exact multiset permutation unrank result violates the stated contract at duplicate block normalization.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The duplicate block normalization step uses math.factorial(sum(counts.values())) instead of ways(counts).

VERIFIED REPAIR

Use ways(counts) at the duplicate block normalization step.

Unsuccessful approach: The partial repair max(1,ways(counts)//2) still violates the duplicate block normalization invariant.

Case contract

Input [sorted multiset of small integer symbols, rank], valid zero-based lexicographic rank among distinct permutations.

Why this case matters

Exact discrete arithmetic with observable algorithmic state; no floating point approximation is used.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    items,r=x
    from collections import Counter
    counts=Counter(items);out=[]
    def ways(c):
     v=math.factorial(sum(c.values()))
     for n in c.values():v//=math.factorial(n)
     return v
    for _ in range(len(items)):
     for symbol in sorted(counts):
      if counts[symbol]==0:continue
      counts[symbol]-=1
      block=math.factorial(sum(counts.values()))
      if r<block:
       out.append(symbol)
       break
      r=r-block
      counts[symbol]=counts[symbol]+1
     else:return None
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[0, 1, 1, 2], 3], [1, 0, 1, 2]), ([[0, 0, 1], 1], [0, 1, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 1], 2], [1, 0, 0]), ([[0, 1, 1, 2], 0], [0, 1, 1, 2]), ([[0, 1, 1, 2], 1], [0, 1, 2, 1]), ([[0, 1, 1, 2], 2], [0, 2, 1, 1])], [([[0, 1, 1, 2], 4], [1, 0, 2, 1]), ([[0, 1, 1, 2], 2], [0, 2, 1, 1]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 1, 1], 2], [0, 1, 1, 0]), ([[0, 0, 1, 1], 3], [1, 0, 0, 1]), ([[0, 0, 1, 1], 4], [1, 0, 1, 0]), ([[0, 0, 1, 1], 5], [1, 1, 0, 0])], [([[0, 1, 1, 2], 5], [1, 1, 0, 2]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 0, 1, 2], 13], [1, 0, 0, 2, 0]), ([[0, 0, 0, 1, 2], 14], [1, 0, 2, 0, 0]), ([[0, 0, 0, 1, 2], 15], [1, 2, 0, 0, 0]), ([[0, 0, 0, 1, 2], 16], [2, 0, 0, 0, 1]), ([[0, 0, 0, 1, 2], 17], [2, 0, 0, 1, 0])], [([[0, 1, 1, 2], 6], [1, 1, 2, 0]), ([[0, 1, 1, 2], 8], [1, 2, 1, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 1, 2, 3], 10], [1, 3, 0, 2]), ([[0, 1, 2, 3], 11], [1, 3, 2, 0]), ([[0, 1, 2, 3], 12], [2, 0, 1, 3]), ([[0, 1, 2, 3], 13], [2, 0, 3, 1])], [([[0, 1, 1, 2], 7], [1, 2, 0, 1]), ([[0, 1, 1, 2], 11], [2, 1, 1, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[1, 1, 2, 2, 3], 3], [1, 2, 1, 2, 3]), ([[1, 1, 2, 2, 3], 4], [1, 2, 1, 3, 2]), ([[1, 1, 2, 2, 3], 5], [1, 2, 2, 1, 3]), ([[1, 1, 2, 2, 3], 6], [1, 2, 2, 3, 1])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
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
explicit oracle 0	`None`	`[1, 0, 1, 2]`	Failed
explicit oracle 1	`[0, 1, 0]`	`[0, 1, 0]`	Passed
explicit oracle 2	`[0, 0, 1]`	`[0, 0, 1]`	Passed
explicit oracle 3	`[2, 1, 3, 2, 1]`	`[3, 2, 2, 1, 1]`	Failed
explicit oracle 4	`[1, 0, 0]`	`[1, 0, 0]`	Passed
explicit oracle 5	`[0, 1, 1, 2]`	`[0, 1, 1, 2]`	Passed
explicit oracle 6	`[0, 1, 2, 1]`	`[0, 1, 2, 1]`	Passed
explicit oracle 7	`[0, 2, 1, 1]`	`[0, 2, 1, 1]`	Passed

SHA-256 / 438f0f4c89d14930d780f2a0eaae51b222366ca52ffceb3c32f683de6cf4b830

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    items,r=x
    from collections import Counter
    counts=Counter(items);out=[]
    def ways(c):
     v=math.factorial(sum(c.values()))
     for n in c.values():v//=math.factorial(n)
     return v
    for _ in range(len(items)):
     for symbol in sorted(counts):
      if counts[symbol]==0:continue
      counts[symbol]-=1
      block=max(1,ways(counts)//2)
      if r<block:
       out.append(symbol)
       break
      r=r-block
      counts[symbol]=counts[symbol]+1
     else:return None
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[0, 1, 1, 2], 3], [1, 0, 1, 2]), ([[0, 0, 1], 1], [0, 1, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 1], 2], [1, 0, 0]), ([[0, 1, 1, 2], 0], [0, 1, 1, 2]), ([[0, 1, 1, 2], 1], [0, 1, 2, 1]), ([[0, 1, 1, 2], 2], [0, 2, 1, 1])], [([[0, 1, 1, 2], 4], [1, 0, 2, 1]), ([[0, 1, 1, 2], 2], [0, 2, 1, 1]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 1, 1], 2], [0, 1, 1, 0]), ([[0, 0, 1, 1], 3], [1, 0, 0, 1]), ([[0, 0, 1, 1], 4], [1, 0, 1, 0]), ([[0, 0, 1, 1], 5], [1, 1, 0, 0])], [([[0, 1, 1, 2], 5], [1, 1, 0, 2]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 0, 1, 2], 13], [1, 0, 0, 2, 0]), ([[0, 0, 0, 1, 2], 14], [1, 0, 2, 0, 0]), ([[0, 0, 0, 1, 2], 15], [1, 2, 0, 0, 0]), ([[0, 0, 0, 1, 2], 16], [2, 0, 0, 0, 1]), ([[0, 0, 0, 1, 2], 17], [2, 0, 0, 1, 0])], [([[0, 1, 1, 2], 6], [1, 1, 2, 0]), ([[0, 1, 1, 2], 8], [1, 2, 1, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 1, 2, 3], 10], [1, 3, 0, 2]), ([[0, 1, 2, 3], 11], [1, 3, 2, 0]), ([[0, 1, 2, 3], 12], [2, 0, 1, 3]), ([[0, 1, 2, 3], 13], [2, 0, 3, 1])], [([[0, 1, 1, 2], 7], [1, 2, 0, 1]), ([[0, 1, 1, 2], 11], [2, 1, 1, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[1, 1, 2, 2, 3], 3], [1, 2, 1, 2, 3]), ([[1, 1, 2, 2, 3], 4], [1, 2, 1, 3, 2]), ([[1, 1, 2, 2, 3], 5], [1, 2, 2, 1, 3]), ([[1, 1, 2, 2, 3], 6], [1, 2, 2, 3, 1])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
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
explicit oracle 0	`[1, 2, 0, 1]`	`[1, 0, 1, 2]`	Failed
explicit oracle 1	`[1, 0, 0]`	`[0, 1, 0]`	Failed
explicit oracle 2	`[0, 0, 1]`	`[0, 0, 1]`	Passed
explicit oracle 3	`None`	`[3, 2, 2, 1, 1]`	Failed
explicit oracle 4	`None`	`[1, 0, 0]`	Failed
explicit oracle 5	`[0, 1, 1, 2]`	`[0, 1, 1, 2]`	Passed
explicit oracle 6	`[1, 0, 1, 2]`	`[0, 1, 2, 1]`	Failed
explicit oracle 7	`[1, 1, 0, 2]`	`[0, 2, 1, 1]`	Failed

SHA-256 / cd3c13ccd13781422b2e9531612fecba9a67e193d860df296cf8652f3ffac69a

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    items,r=x
    from collections import Counter
    counts=Counter(items);out=[]
    def ways(c):
     v=math.factorial(sum(c.values()))
     for n in c.values():v//=math.factorial(n)
     return v
    for _ in range(len(items)):
     for symbol in sorted(counts):
      if counts[symbol]==0:continue
      counts[symbol]-=1
      block=ways(counts)
      if r<block:
       out.append(symbol)
       break
      r=r-block
      counts[symbol]=counts[symbol]+1
     else:return None
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[0, 1, 1, 2], 3], [1, 0, 1, 2]), ([[0, 0, 1], 1], [0, 1, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 1], 2], [1, 0, 0]), ([[0, 1, 1, 2], 0], [0, 1, 1, 2]), ([[0, 1, 1, 2], 1], [0, 1, 2, 1]), ([[0, 1, 1, 2], 2], [0, 2, 1, 1])], [([[0, 1, 1, 2], 4], [1, 0, 2, 1]), ([[0, 1, 1, 2], 2], [0, 2, 1, 1]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 1, 1], 2], [0, 1, 1, 0]), ([[0, 0, 1, 1], 3], [1, 0, 0, 1]), ([[0, 0, 1, 1], 4], [1, 0, 1, 0]), ([[0, 0, 1, 1], 5], [1, 1, 0, 0])], [([[0, 1, 1, 2], 5], [1, 1, 0, 2]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 0, 1, 2], 13], [1, 0, 0, 2, 0]), ([[0, 0, 0, 1, 2], 14], [1, 0, 2, 0, 0]), ([[0, 0, 0, 1, 2], 15], [1, 2, 0, 0, 0]), ([[0, 0, 0, 1, 2], 16], [2, 0, 0, 0, 1]), ([[0, 0, 0, 1, 2], 17], [2, 0, 0, 1, 0])], [([[0, 1, 1, 2], 6], [1, 1, 2, 0]), ([[0, 1, 1, 2], 8], [1, 2, 1, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 1, 2, 3], 10], [1, 3, 0, 2]), ([[0, 1, 2, 3], 11], [1, 3, 2, 0]), ([[0, 1, 2, 3], 12], [2, 0, 1, 3]), ([[0, 1, 2, 3], 13], [2, 0, 3, 1])], [([[0, 1, 1, 2], 7], [1, 2, 0, 1]), ([[0, 1, 1, 2], 11], [2, 1, 1, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[1, 1, 2, 2, 3], 3], [1, 2, 1, 2, 3]), ([[1, 1, 2, 2, 3], 4], [1, 2, 1, 3, 2]), ([[1, 1, 2, 2, 3], 5], [1, 2, 2, 1, 3]), ([[1, 1, 2, 2, 3], 6], [1, 2, 2, 3, 1])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
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
explicit oracle 0	`[1, 0, 1, 2]`	`[1, 0, 1, 2]`	Passed
explicit oracle 1	`[0, 1, 0]`	`[0, 1, 0]`	Passed
explicit oracle 2	`[0, 0, 1]`	`[0, 0, 1]`	Passed
explicit oracle 3	`[3, 2, 2, 1, 1]`	`[3, 2, 2, 1, 1]`	Passed
explicit oracle 4	`[1, 0, 0]`	`[1, 0, 0]`	Passed
explicit oracle 5	`[0, 1, 1, 2]`	`[0, 1, 1, 2]`	Passed
explicit oracle 6	`[0, 1, 2, 1]`	`[0, 1, 2, 1]`	Passed
explicit oracle 7	`[0, 2, 1, 1]`	`[0, 2, 1, 1]`	Passed

SHA-256 / 4a91e4aa7660771e843040f3b8662844a66e4a1468ddc5e09c032721109a671e

Verification & scope

A deterministic bounded teaching model. Inputs are restricted to the explicit contract; this is not a production algebra library. 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 2026-09-29T14:39:24.622046+00:00.

Case digest / 20ead7d2e77f870373935d8333b0bec721af6a9186728daaea20a83473aba983