Description
Calculate symmetric and smaller binomial-coefficient indices.
Combinatorial checks often evaluate necessary conditions or one algebraic component. Parity can be combined from supplied signs, a divisibility test can constrain an orbit size, and prefix sums can enforce a strict positivity condition. Passing one condition does not construct or prove the existence of the full combinatorial object.
Calculate symmetric and smaller binomial-coefficient indices.
When to use this tool
Use Binomial Coefficient Symmetry Index when you already have n, k and need to calculate symmetric and smaller binomial-coefficient indices.
How to use the tool and how it works
- Enter n, k. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read symmetricIndex, reducedIndex. Use the output convention below.
Require safe whole n>=0 and k in [0,n]; return n-k and min(k,n-k).1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| n | N | Required. Safe integer; minimum 0. |
| k | K | Required. Safe integer; minimum 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| symmetricIndex | Symmetric Index | Required. Safe integer. |
| reducedIndex | min(k,n-k); not the binomial coefficient itself. | Required. Safe integer. |
{
"input": {
"n": 10,
"k": 7
},
"output": {
"symmetricIndex": 3,
"reducedIndex": 3
}
}Limitations and assumptions
- No binomial coefficient, probability, combinatorial object or application-specific count is computed.
- 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
Use a complete construction, enumeration or proof for the mathematical question. Keep integer ranges, empty-input conventions and strict inequalities explicit when incorporating these local checks into a larger algorithm.
References
Similar or alternative tools
- Orbit Size Divisibility Remainder
Check divisibility of a group order by a proposed orbit size.
- Permutation Inversion Parity Compose
Compose two supplied permutation parity flags.