Index Permutation Cycle Table
Decompose an index permutation into disjoint cycles, including fixed points and starting each cycle at its smallest unseen index.
Description
Decompose an index permutation into disjoint cycles, including fixed points and starting each cycle at its smallest unseen index.
Disjoint cycles partition a finite index permutation, including its fixed points.
When to use this tool
Decompose an index permutation into disjoint cycles, including fixed points and starting each cycle at its smallest unseen index.
- Prepare the supplied permutation fields for this operation.
- Compare the complete example with the representation used by your own input before carrying the result into another step.
How to use the tool and how it works
- Enter permutation. Use the exact text syntax, units and index conventions explained below.
- Run the tool. If input validation fails, correct the supplied fields before interpreting any result.
- Read result.cycles. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.
Validate the permutation, scan indexes ascending and follow each unseen index until its cycle closes. Return the cycle lists, each beginning at its smallest index.1
| Input field | Meaning and representation | Accepted input |
|---|---|---|
| permutation | permutation. | List of finite numbers; at most 128 entries |
| Output field | Interpretation |
|---|---|
| result | Object containing the named outputs below. |
| result.cycles | The cycles list in the ordering and representation described above. |
{
"input": {
"permutation": [
1,
0,
2
]
},
"output": {
"result": {
"cycles": [
[
0,
1
],
[
2
]
]
}
}
}Limitations and assumptions
- The start is not repeated at the cycle end. No cycle-notation string or external value labels are produced.
- Except in the diagnostic validator, a permutation must contain each integer from zero through length minus one exactly once. Lists contain at most 128 entries.
- All declared input fields are required and unexpected top-level fields are rejected. Numbers must be finite; lists are limited to 128 entries and ordinary text fields to 8,192 UTF-16 code units, with tighter limits where stated.
- JSON text inputs must parse within bounded nesting and size limits. Numeric outputs outside the safe finite range are rejected; large exact integer results use decimal strings. Indexes and half-open spans are zero-based unless a rank is explicitly described as one-based.
Alternative or Complementary methods
Keep the index-to-image convention and composition direction explicit. A rank, cycle table, reorder-index list and inverse map encode different operations even when their entries are all integers.
References
Similar or alternative tools
- Permutation Unrestricted Swap Distance
Compute minimum unrestricted swap distance between two index permutations from relative cycle count; adjacent-swap distance is different.
- Permutation Cycle Notation Formatter
Format canonical zero-based disjoint permutation cycles with an explicit fixed-point display policy; no one-based convention implied.