Description
Check whether a requested fixed-point count leaves a possible derangement size.
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 whether a requested fixed-point count leaves a possible derangement size.
When to use this tool
Use Permutation Fixed Point Free 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 remaining, feasible. Inspect feasible before interpreting the associated values.
Check whether a requested fixed-point count leaves a possible derangement size. Require valid whole size and fixedPoints in [0,size]; remaining = size-fixedPoints, feasible means remaining != 1.1
| 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 |
|---|---|---|
| remaining | Budget minus use under the stated direction, retaining negative deficits where allowed. | Required. Safe integer. |
| feasible | Whether the stated local constraints hold; this need not establish a valid complete object or procedure. | Required. Boolean. |
{
"input": {
"size": 5,
"fixedPoints": 4
},
"output": {
"remaining": 1,
"feasible": false
}
}Limitations and assumptions
- Feasibility is cardinality-only for unrestricted permutations; no actual permutation or constrained assignment 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
- Involution Single Fixed Point Budget
Check an involution's remaining pair budget.
- Permutation Prefix Forced Fixed Count
Count forced fixed points in a permutation prefix.