C#，欧拉数（Eulerian Number）的算法与源代码

1 欧拉数

欧拉数特指 Eulerian Number，不同于 Euler numbers，Euler's number 哦。

组合数学中，欧拉数（Eulerian Number）是从1到n中正好满足m个元素大于前一个元素（具有m个“上升”的排列）条件的排列个数。定义为：

计算公式：

2 文本格式

using System;

namespace Legalsoft.Truffer.Algorithm
{
   public static partial class Number_Sequence
   {
       /// <summary>
       /// 欧拉数（递归）算法
       /// </summary>
       /// <param name="n"></param>
       /// <param name="m"></param>
       /// <returns></returns>
       public static int Eulerian_Number(int n, int m)
       {
           if (m >= n || n == 0)
           {
               return 0;
           }
           if (m == 0)
           {
               return 1;
           }
           int a = (n - m) * Eulerian_Number(n - 1, m - 1);
           int b = (m + 1) * Eulerian_Number(n - 1, m);
           return (a + b);
       }

       /// <summary>
       /// 欧拉数（非递归）算法
       /// </summary>
       /// <param name="n"></param>
       /// <param name="m"></param>
       /// <returns></returns>
       public static int Eulerian_Number_Second(int n, int m)
       {
           int[,] dp = new int[n + 1, m + 1];
           for (int i = 1; i <= n; i++)
           {
               for (int j = 0; j <= m; j++)
               {
                   if (i > j)
                   {
                       if (j == 0)
                       {
                           dp[i, j] = 1;
                       }
                       else
                       {
                           dp[i, j] = ((i - j) * dp[i - 1, j - 1]) + ((j + 1) * dp[i - 1, j]);
                       }
                   }
               }
           }
           return dp[n, m];
       }
   }
}

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3 代码格式

using System;namespace Legalsoft.Truffer.Algorithm
{public static partial class Number_Sequence{/// <summary>/// 欧拉数（递归）算法/// </summary>/// <param name="n"></param>/// <param name="m"></param>/// <returns></returns>public static int Eulerian_Number(int n, int m){if (m >= n || n == 0){return 0;}if (m == 0){return 1;}int a = (n - m) * Eulerian_Number(n - 1, m - 1);int b = (m + 1) * Eulerian_Number(n - 1, m);return (a + b);}/// <summary>/// 欧拉数（非递归）算法/// </summary>/// <param name="n"></param>/// <param name="m"></param>/// <returns></returns>public static int Eulerian_Number_Second(int n, int m){int[,] dp = new int[n + 1, m + 1];for (int i = 1; i <= n; i++){for (int j = 0; j <= m; j++){if (i > j){if (j == 0){dp[i, j] = 1;}else{dp[i, j] = ((i - j) * dp[i - 1, j - 1]) + ((j + 1) * dp[i - 1, j]);}}}}return dp[n, m];}}
}

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