Description
Return orbit size from group and stabilizer cardinalities.
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 orbit size from group and stabilizer cardinalities.
When to use this tool
Use Combinatorics: Necklace Rotation Stabilizer Quotient when checking the stated partition, permutation or rotation/reflection convention.
How to use the tool and how it works
- Enter rotationGroupSize, stabilizerSize. 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 orbit size from group and stabilizer cardinalities. Require bounded positive whole sizes, stabilizer <= group and exact divisibility; orbitSize = rotationGroupSize/stabilizerSize.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| rotationGroupSize | Rotation Group Size | Required. Safe integer. |
| stabilizerSize | Stabilizer Size | Required. Safe integer. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Returned calculated value or nested object under the stated output convention. | Required. Object. |
| result.orbitSize | Orbit Size | Calculated result member: number. |
{
"input": {
"rotationGroupSize": 12,
"stabilizerSize": 3
},
"output": {
"result": {
"orbitSize": 4
}
}
}Limitations and assumptions
- This assumes the supplied stabilizer is genuine; no necklace, rotation symmetry or group action is verified.
- 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
- Necklace Rotation Fixed Position Count
Return position cycles induced by a rotation.
- Necklace Period Divisor Flag
Check whether a proposed period divides a word length.