Description
Return permutation sign from size and cycle count.
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.
Return permutation sign from size and cycle count.
When to use this tool
Use Combinatorics: Permutation Cycle Parity From Count when checking the stated partition, permutation or rotation/reflection convention.
How to use the tool and how it works
- Enter size, cycleCount. 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 permutation sign from size and cycle count. Require size 0..1000000 and cycleCount zero only for empty size, otherwise 1..size; odd tests (size-cycles) modulo two and sign is plus/minus one.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| size | Size | Required. Safe integer. |
| cycleCount | Number of cycles including fixed-point one-cycles; zero only for empty permutation. | Required. Safe integer. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Returned calculated value or nested object under the stated output convention. | Required. Object. |
| result.odd | Odd | Calculated result member: boolean. |
| result.sign | Sign | Calculated result member: number. |
{
"input": {
"size": 5,
"cycleCount": 2
},
"output": {
"result": {
"odd": true,
"sign": -1
}
}
}Limitations and assumptions
- Cycle count includes fixed-point cycles; no actual permutation or cycle decomposition is inspected.
- 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
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 Transposition Count Parity
Return sign and parity of a transposition count.
- Permutation Cycle Split Sign
Return relative sign from a supplied cycle-count change.