Description
Check an involution's remaining pair budget.
Partitions, permutations and circular words use distinct encodings and symmetry conventions. Parity, block bounds and orbit-size identities can follow from supplied counts without verifying a complete object or identifying its actual symmetry.
Check an involution's remaining pair budget.
When to use this tool
Use Involution Single Fixed Point Budget when checking the stated partition, permutation or rotation/reflection convention.
How to use the tool and how it works
- Enter size, fixedPoints. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read valid, pairs. Inspect valid before interpreting the associated values.
Check an involution's remaining pair budget. Require valid whole size and fixedPoints within it; rest = size-fixed, valid requires even rest, pairs = floor(rest/2).1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| size | Size | Required. Safe integer; minimum 0. |
| fixedPoints | Fixed Points | Required. Safe integer; minimum 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| valid | Whether this local input/result condition holds; inspect the limitations before treating it as full validation. | Required. Boolean. |
| pairs | Truncated half of the nonfixed remainder; inspect valid for odd remainders. | Required. Safe integer. |
{
"input": {
"size": 8,
"fixedPoints": 2
},
"output": {
"valid": true,
"pairs": 3
}
}Limitations and assumptions
- When valid is false, pairs is a truncated placeholder; no actual involution or graph matching is constructed.
- 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
Validate the complete combinatorial object and its action/labeling convention before applying whole-object enumeration, ranking or symmetry conclusions.
References
Similar or alternative tools
- Permutation Fixed Point Free Budget
Check whether a requested fixed-point count leaves a possible derangement size.
- Permutation Prefix Forced Fixed Count
Count forced fixed points in a permutation prefix.