Index Permutation Order
Return exact permutation order as the least common multiple of cycle lengths; the empty permutation has order one.
Description
Return exact permutation order as the least common multiple of cycle lengths; the empty permutation has order one.
A permutation's order is the number of repeated applications needed to return to identity. It is the least common multiple of cycle lengths.
When to use this tool
Return exact permutation order as the least common multiple of cycle lengths; the empty permutation has order one.
- 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.order. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.
Validate and find cycles, then accumulate their least common multiple using exact integer greatest-common-divisor arithmetic. Return order as decimal text, beginning from one.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.order | The least common multiple of cycle lengths as exact decimal integer text; one for the empty permutation. |
{
"input": {
"permutation": [
1,
0,
3,
4,
2
]
},
"output": {
"result": {
"order": "6"
}
}
}Limitations and assumptions
- The empty permutation has order one. No repeated application or full group enumeration is performed.
- 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
- Index Permutation Composition
Compose index permutations by applying first then second to each index; composition convention is explicit and operand lengths must agree.
- Index Permutation Cycle Table
Decompose an index permutation into disjoint cycles, including fixed points and starting each cycle at its smallest unseen index.