Class Combinatorics

java.lang.Object
cloud.opencode.base.math.combinatorics.Combinatorics
public final class Combinatorics extends Object
Combinatorial mathematics utility methods. 组合数学工具方法集合

Provides binomial coefficients, permutations, Catalan numbers, Stirling numbers of the second kind, Bell numbers, and derangements. All methods are stateless and thread-safe.

提供二项式系数、排列数、Catalan 数、第二类 Stirling 数、Bell 数和错排数。 所有方法无状态且线程安全。

Since:
JDK 25, opencode-base-math V1.0.3
Author:
Leon Soo www.LeonSoo.com
See Also:

  • Method Summary

    Modifier and Type
    Method
    Description
    static long
    bellNumber(int n)
    Computes the n-th Bell number (sum of Stirling numbers of the second kind).
    static long
    binomial(int n, int k)
    Computes the binomial coefficient C(n, k) = n!
    static BigInteger
    binomialBig(int n, int k)
    Computes the binomial coefficient C(n, k) with arbitrary precision.
    static BigInteger
    catalanBig(int n)
    Computes the n-th Catalan number with arbitrary precision.
    static long
    catalanNumber(int n)
    Computes the n-th Catalan number: C(2n, n) / (n + 1).
    static long
    derangements(int n)
    Computes the number of derangements (subfactorial) !
    static long
    permutation(int n, int k)
    Computes the permutation P(n, k) = n!
    static BigInteger
    permutationBig(int n, int k)
    Computes the permutation P(n, k) with arbitrary precision.
    static long
    stirlingSecond(int n, int k)
    Computes the Stirling number of the second kind S(n, k).

    Methods inherited from class Object

    clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

  • Method Details

    • binomial

      public static long binomial(int n, int k)
      Computes the binomial coefficient C(n, k) = n! / (k! * (n-k)!). 计算二项式系数 C(n, k) = n! / (k! * (n-k)!)

      Uses the multiplicative formula to avoid intermediate overflow where possible. Throws MathException if the result overflows long.

      使用乘法公式尽量避免中间溢出。如果结果超出 long 范围则抛出 MathException。

      Parameters:
      n - total elements, must be ≥ 0 / 总元素数，必须 ≥ 0
      k - chosen elements, must satisfy 0 ≤ k ≤ n / 选择元素数，必须满足 0 ≤ k ≤ n
      Returns:
      C(n, k) / 二项式系数
      Throws:
      IllegalArgumentException - if n < 0, k < 0, or k > n
      MathException - if result overflows long / 如果结果溢出 long

    • binomialBig

      public static BigInteger binomialBig(int n, int k)
      Computes the binomial coefficient C(n, k) with arbitrary precision. 使用任意精度计算二项式系数 C(n, k)
      Parameters:
      n - total elements, must be ≥ 0 / 总元素数，必须 ≥ 0
      k - chosen elements, must satisfy 0 ≤ k ≤ n / 选择元素数，必须满足 0 ≤ k ≤ n
      Returns:
      C(n, k) as BigInteger / BigInteger 形式的二项式系数
      Throws:
      IllegalArgumentException - if n < 0, k < 0, or k > n

    • permutation

      public static long permutation(int n, int k)
      Computes the permutation P(n, k) = n! / (n-k)!. 计算排列数 P(n, k) = n! / (n-k)!
      Parameters:
      n - total elements, must be ≥ 0 / 总元素数，必须 ≥ 0
      k - chosen elements, must satisfy 0 ≤ k ≤ n / 选择元素数，必须满足 0 ≤ k ≤ n
      Returns:
      P(n, k) / 排列数
      Throws:
      IllegalArgumentException - if n < 0, k < 0, or k > n
      MathException - if result overflows long / 如果结果溢出 long

    • permutationBig

      public static BigInteger permutationBig(int n, int k)
      Computes the permutation P(n, k) with arbitrary precision. 使用任意精度计算排列数 P(n, k)
      Parameters:
      n - total elements, must be ≥ 0 / 总元素数，必须 ≥ 0
      k - chosen elements, must satisfy 0 ≤ k ≤ n / 选择元素数，必须满足 0 ≤ k ≤ n
      Returns:
      P(n, k) as BigInteger / BigInteger 形式的排列数
      Throws:
      IllegalArgumentException - if n < 0, k < 0, or k > n

    • catalanNumber

      public static long catalanNumber(int n)
      Computes the n-th Catalan number: C(2n, n) / (n + 1). 计算第 n 个 Catalan 数：C(2n, n) / (n + 1)
      Parameters:
      n - the index, must be ≥ 0 / 索引，必须 ≥ 0
      Returns:
      the n-th Catalan number / 第 n 个 Catalan 数
      Throws:
      IllegalArgumentException - if n < 0
      MathException - if result overflows long / 如果结果溢出 long

    • catalanBig

      public static BigInteger catalanBig(int n)
      Computes the n-th Catalan number with arbitrary precision. 使用任意精度计算第 n 个 Catalan 数
      Parameters:
      n - the index, must be ≥ 0 / 索引，必须 ≥ 0
      Returns:
      the n-th Catalan number as BigInteger / BigInteger 形式的 Catalan 数
      Throws:
      IllegalArgumentException - if n < 0

    • stirlingSecond

      public static long stirlingSecond(int n, int k)
      Computes the Stirling number of the second kind S(n, k). 计算第二类 Stirling 数 S(n, k)

      Uses the recurrence S(n, k) = k * S(n-1, k) + S(n-1, k-1), with base cases S(n, 0) = 0 (n > 0), S(0, 0) = 1, S(n, n) = 1.

      使用递推关系 S(n, k) = k * S(n-1, k) + S(n-1, k-1)， 边界条件 S(n, 0) = 0 (n > 0)，S(0, 0) = 1，S(n, n) = 1。

      Parameters:
      n - number of elements, must be ≥ 0 / 元素数，必须 ≥ 0
      k - number of non-empty subsets, must satisfy 0 ≤ k ≤ n / 非空子集数，必须满足 0 ≤ k ≤ n
      Returns:
      S(n, k) / 第二类 Stirling 数
      Throws:
      IllegalArgumentException - if n < 0, k < 0, or k > n

    • bellNumber

      public static long bellNumber(int n)
      Computes the n-th Bell number (sum of Stirling numbers of the second kind). 计算第 n 个 Bell 数（第二类 Stirling 数之和）

      B(n) = sum_{k=0}^{n} S(n, k).

      Parameters:
      n - the index, must be ≥ 0 / 索引，必须 ≥ 0
      Returns:
      the n-th Bell number / 第 n 个 Bell 数
      Throws:
      IllegalArgumentException - if n < 0
      MathException - if result overflows long / 如果结果溢出 long

    • derangements

      public static long derangements(int n)
      Computes the number of derangements (subfactorial) !n. 计算错排数（子阶乘）!n

      A derangement is a permutation where no element appears in its original position. Uses the recurrence !n = (n-1) * (!(n-1) + !(n-2)).

      错排是指没有元素出现在其原始位置的排列。 使用递推关系 !n = (n-1) * (!(n-1) + !(n-2))。

      Parameters:
      n - the number of elements, must be ≥ 0 / 元素数量，必须 ≥ 0
      Returns:
      !n (the number of derangements) / 错排数
      Throws:
      IllegalArgumentException - if n < 0
      MathException - if result overflows long / 如果结果溢出 long