Permutation Cycle Notation Formatter
Format canonical zero-based disjoint permutation cycles with an explicit fixed-point display policy; no one-based convention implied.
Description
Format canonical zero-based disjoint permutation cycles with an explicit fixed-point display policy; no one-based convention implied.
Cycle notation follows repeated application of an index permutation. Fixed points may be shown or omitted by policy.
When to use this tool
Format canonical zero-based disjoint permutation cycles with an explicit fixed-point display policy; no one-based convention implied.
- Prepare the supplied permutation, includeFixed 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, includeFixed. 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.notation. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.
Find cycles starting at the smallest unseen index. Optionally remove one-element cycles, then format each as parenthesized space-separated indexes with no separator between cycles.1
| Input field | Meaning and representation | Accepted input |
|---|---|---|
| permutation | permutation. | List of finite numbers; at most 128 entries |
| includeFixed | Explicit choice for include fixed; supply a boolean rather than a text label. | true or false |
| Output field | Interpretation |
|---|---|
| result | Object containing the named outputs below. |
| result.notation | The notation value under the stated formula or text convention. |
{
"input": {
"permutation": [
1,
0,
2
],
"includeFixed": true
},
"output": {
"result": {
"notation": "(0 1)(2)"
}
}
}Limitations and assumptions
- When fixed points are omitted, external size is needed to reconstruct the whole permutation. No arbitrary symbol labels or alternate cycle syntax is generated.
- 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 Cycle Notation Parser
Parse zero-based disjoint cycle notation on an explicit index universe; overlapping cycles are rejected instead of interpreted as cycle products.
- Index Permutation Cycle Table
Decompose an index permutation into disjoint cycles, including fixed points and starting each cycle at its smallest unseen index.