Description
Return position cycles induced by a rotation.
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 position cycles induced by a rotation.
When to use this tool
Use Necklace Rotation Fixed Position Count when checking the stated partition, permutation or rotation/reflection convention.
How to use the tool and how it works
- Enter length, rotation. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read cycles. Use the output convention below.
Return position cycles induced by a rotation. Require bounded positive whole length; cycles = gcd(length,rotation modulo length) using nonnegative modulo.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| length | Length | Required. Safe integer; minimum 1; maximum 1000000. |
| rotation | Rotation | Required. Safe integer. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| cycles | Cycles | Required. Safe integer. |
{
"input": {
"length": 12,
"rotation": 8
},
"output": {
"cycles": 4
}
}Limitations and assumptions
- This counts index cycles, not distinct necklaces or fixed colorings without an additional alphabet-size model.
- 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
- Combinatorics: Necklace Rotation Stabilizer Quotient
Return orbit size from group and stabilizer cardinalities.
- Necklace Period Divisor Flag
Check whether a proposed period divides a word length.