Description
Return relative sign from a supplied cycle-count change.
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 relative sign from a supplied cycle-count change.
When to use this tool
Use Permutation Cycle Split Sign when checking the stated partition, permutation or rotation/reflection convention.
How to use the tool and how it works
- Enter originalCycleCount, resultingCycleCount. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read cycleCountChange, relativeSign. Use the output convention below.
Return relative sign from a supplied cycle-count change. Require nonnegative whole original/resulting counts; change = resulting-original, relativeSign = +1 for even change and -1 for odd.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| originalCycleCount | Original Cycle Count | Required. Safe integer; minimum 0. |
| resultingCycleCount | Resulting Cycle Count | Required. Safe integer; minimum 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| cycleCountChange | Cycle Count Change | Required. Safe integer. |
| relativeSign | Relative Sign | Required. Safe integer. |
{
"input": {
"originalCycleCount": 1,
"resultingCycleCount": 2
},
"output": {
"cycleCountChange": 1,
"relativeSign": -1
}
}Limitations and assumptions
- The parity identity assumes equal permutation size; supplied counts alone do not verify compatible permutations or a particular cycle operation.
- 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.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
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
- Combinatorics: Permutation Cycle Parity From Count
Return permutation sign from size and cycle count.
- Permutation Fixed Point Free Budget
Check whether a requested fixed-point count leaves a possible derangement size.