combinatorics-calculator

运用模运算计算组合数值

SKILL.md

--- frontmatter

name: combinatorics-calculator
description: Calculate combinatorial values with modular arithmetic
allowed-tools:
  - Read
  - Write
  - Grep
  - Glob
  - Edit

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

json

{
  "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

json

{
  "type": "object",
  "properties": {
    "success": { "type": "boolean" },
    "result": { "type": "integer" },
    "code": { "type": "string" },
    "formula": { "type": "string" }
  },
  "required": ["success"]
}