Index Permutation Sign
Compute permutation parity from its disjoint-cycle count, not from numeric value signs; empty and identity permutations are even.
Description
Compute permutation parity from its disjoint-cycle count, not from numeric value signs; empty and identity permutations are even.
Permutation sign records whether its parity is even or odd. Cycle counts provide a direct calculation.
When to use this tool
Compute permutation parity from its disjoint-cycle count, not from numeric value signs; empty and identity permutations are even.
- 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.sign, result.odd. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.
Validate and decompose the permutation. Let k be length minus cycle count; sign is -1 for odd k and +1 for even k, with an accompanying odd flag.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.sign | The sign value under the stated formula or text convention. |
| result.odd | Flag for odd under the exact condition described above. |
{
"input": {
"permutation": [
1,
0,
2
]
},
"output": {
"result": {
"sign": -1,
"odd": true
}
}
}Limitations and assumptions
- This is permutation parity rather than the signs of its numeric entries. The empty permutation has sign +1.
- 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.