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"]
}