Combinatorics Calculator (nCr/nPr/multinomial) mechanism

Model scope

The three supplied formulas are standard and pass the four checks above, including the n=0,r=0 boundary. The overall specification cannot be accepted as complete because it claims permutations with repetition, Catalan numbers, and Stirling numbers without defining their meanings or algorithms, and it omits essential input constraints.

• The stated formulas correctly define ordinary combinations, ordinary permutations, and multinomial coefficients, subject to integer-domain constraints.
• 'Permutations with repetition' is not defined and is ambiguous: it may mean ordered selections with replacement (n^r) or permutations of a multiset (n!/(k1!k2!...)).
• Catalan numbers are claimed but no Catalan formula or algorithm is provided; a standard formula is C_n=(1/(n+1))·binomial(2n,n).
• Stirling numbers are claimed but no type (first or second kind), recurrence, or formula is provided.
• The multinomial formula requires nonnegative integer parts k1,...,km whose sum equals n; this required constraint is not stated.
• nCr requires integers n>=0 and 0<=r<=n; nPr requires integers n>=0 and 0<=r<=n. These domain constraints are omitted.

Units and conventions

• n, r, and every partition entry k_i are dimensionless, nonnegative integers.
• Factorials are defined only for nonnegative integers, with 0!=1.
• For a multinomial input given only as a partition list, n is assumed to be the exact sum of the list entries.
• Outputs are exact integer counts, not floating-point approximations.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "operation": "nCr",
    "n": 0,
    "r": 0
  },
  "expected_outputs": {
    "coefficient": 1
  },
  "working": "0!/(0!·(0-0)!) = 0!/(0!·0!) = 1/(1·1) = 1. This checks the zero boundary and uses 0!=1."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The specification exposes an undefined factorial domain at r>n and does not state required rejection behavior. It also lists permutations with repetition without giving its formula or an explicit convention selector, allowing the same nPr-looking input to produce different results.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json