Description
Return one inclusion-exclusion coefficient under an empty-subset policy.
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 inclusion-exclusion coefficient under an empty-subset policy.
When to use this tool
Use Inclusion Exclusion Single Subset Sign when checking the stated recurrence, coefficient-ratio or inclusion-exclusion term.
How to use the tool and how it works
- Enter subsetSize, includeEmpty. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read coefficient. Use the output convention below.
Return one inclusion-exclusion coefficient under an empty-subset policy. Require valid whole subsetSize; positive size gives +1 for odd and -1 for even, while zero gives one only when includeEmpty.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| subsetSize | Subset Size | Required. Safe integer; minimum 0. |
| includeEmpty | Include Empty | Required. Boolean. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| coefficient | Odd/even union-term sign for positive size; zero-size coefficient follows includeEmpty explicitly. | Required. Safe integer. |
{
"input": {
"subsetSize": 3,
"includeEmpty": false
},
"output": {
"coefficient": 1
}
}Limitations and assumptions
- The empty coefficient is this helper's explicit convention, not automatically the usual union formula or complement indicator.
- 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
- Combinatorics: Derangement Inclusion Term
Return one exact derangement inclusion-exclusion term.
- Catalan Convolution Single Summand Tool
Compute catalan convolution single summand from explicit bounded inputs.