Description
Return the exact product of multiplicity factorials.
Combinatorial counts depend on whether objects are labeled, order matters, repetitions are permitted and a prefix is already fixed. Local counts, exact integer factors and feasibility bounds answer specific questions and need not enumerate complete objects.
Return the exact product of multiplicity factorials.
When to use this tool
Use Combinatorics: Multiset Multiplicity Factorial Denominator when checking the stated word, set, multiset or composition convention.
How to use the tool and how it works
- Enter multiplicities. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read result. Expand result to inspect the returned fields.
Return the exact product of multiplicity factorials. Require whole multiplicities 0..128 with total <= 128; factorialDenominator = product(n_i!) as decimal text, with empty product one.1
Multinomial counts divide the total factorial by multiplicity factorials; merging categories changes that denominator rather than preserving the arrangement count.2
| Input field | Meaning and units | Accepted input |
|---|---|---|
| multiplicities | Multiplicities | Required. Array of finite number. Implementation limit: 128 items. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Returned calculated value or nested object under the stated output convention. | Required. Object. |
| result.factorialDenominator | Factorial Denominator | Calculated result member: string. |
{
"input": {
"multiplicities": [
2,
3
]
},
"output": {
"result": {
"factorialDenominator": "12"
}
}
}Limitations and assumptions
- This is the denominator in a multiset arrangement count, not the complete multinomial coefficient.
- Outputs are nested inside result. Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use complete objects and an appropriate exact counting, ranking or enumeration method when the local contribution is insufficient.
References
Similar or alternative tools
- Falling Factorial Last Factor Index
Return the last factor of a falling factorial with an empty convention.
- Multinomial Category Merge Factor
Return an exact binomial factor for merging two category counts.