Description
Return one exact derangement inclusion-exclusion term.
Exact combinatorial recurrences distinguish individual summands, adjacent ratios and complete totals. A local arithmetic identity can be correct without proving that its supplied inputs are the intended counts or that a whole enumeration has been completed.
Return one exact derangement inclusion-exclusion term.
When to use this tool
Use Combinatorics: Derangement Inclusion Term when checking the stated recurrence, coefficient-ratio or inclusion-exclusion term.
How to use the tool and how it works
- Enter size, excludedCount. 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 one exact derangement inclusion-exclusion term. Require size 0..128 and excludedCount in [0,size]; signedInclusionTerm = (-1)^excludedCount*size!/excludedCount! as decimal text.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| size | Size | Required. Safe integer. |
| excludedCount | Excluded Count | Required. Safe integer. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Returned calculated value or nested object under the stated output convention. | Required. Object. |
| result.signedInclusionTerm | Signed Inclusion Term | Calculated result member: string. |
{
"input": {
"size": 5,
"excludedCount": 2
},
"output": {
"result": {
"signedInclusionTerm": "60"
}
}
}Limitations and assumptions
- The term is not the full derangement count; all terms must use the same total-size convention.
- 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 the complete recurrence with justified initial conditions and exact integer arithmetic when a full counting result is required.
References
Similar or alternative tools
- Inclusion Exclusion Single Subset Sign
Return one inclusion-exclusion coefficient under an empty-subset policy.
- Combinatorics: Binomial Adjacent Coefficient Ratio Tool
Compute C(n,k+1)/C(n,k) for one valid k<n without building a binomial row.