Description
Return a bead word's rotation orbit and stabilizer sizes.
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 a bead word's rotation orbit and stabilizer sizes.
When to use this tool
Use Necklace Orbit Size when checking the stated partition, permutation or rotation/reflection convention.
How to use the tool and how it works
- Enter beads. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read orbitSize, stabilizerSize. Use the output convention below.
Return a bead word's rotation orbit and stabilizer sizes. Require nonempty beads; find the smallest divisor p of length whose pattern repeats literally, orbitSize = p, stabilizerSize = length/p.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| beads | Beads | Required. Array of text; maximum UTF-16 length 8192; maximum items 256. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| orbitSize | Orbit Size | Required. Safe integer. |
| stabilizerSize | Stabilizer Size | Required. Safe integer. |
{
"input": {
"beads": [
"A",
"B",
"A",
"B"
]
},
"output": {
"orbitSize": 2,
"stabilizerSize": 2
}
}Limitations and assumptions
- This quotients rotations only; reflections, color relabeling and semantic bead equivalence are not included.
- 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.
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
- Circular Arrangement Reflection Orbit Size
Extend a rotation orbit by a supplied reflection-symmetry flag.
- Necklace Period Divisor Flag
Check whether a proposed period divides a word length.