a5c-ai Combinatorics Calculator Skill
Purpose
Calculate combinatorial values with modular arithmetic support for competitive programming applications.
Capabilities
- Factorial and inverse factorial precomputation
- nCr, nPr with modular arithmetic
- Catalan, Stirling, Bell numbers
- Lucas theorem implementation
- Inclusion-exclusion principle application
- Generating functions
Target Processes
- combinatorics-counting
- number-theory-algorithms
- dp-pattern-matching
Combinatorial Functions
Basic Counting
- Factorial: n!
- Permutations: P(n,r) = n!/(n-r)!
- Combinations: C(n,r) = n!/(r!(n-r)!)
Special Numbers
- Catalan numbers
- Stirling numbers (first and second kind)
- Bell numbers
- Derangements
Advanced Techniques
- Lucas theorem (for large n, small p)
- Inclusion-exclusion
- Burnside's lemma
- Generating functions
Input Schema
{
"type": "object",
"properties": {
"operation": {
"type": "string",
"enum": ["nCr", "nPr", "factorial", "catalan", "stirling", "lucas", "precompute"]
},
"n": { "type": "integer" },
"r": { "type": "integer" },
"mod": { "type": "integer" },
"precomputeLimit": { "type": "integer" }
},
"required": ["operation"]
}
Output Schema
{
"type": "object",
"properties": {
"success": { "type": "boolean" },
"result": { "type": "integer" },
"code": { "type": "string" },
"formula": { "type": "string" }
},
"required": ["success"]
}