Hduoj1023【卡特兰数】

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/*Train Problem IITime Limit: 2000/1000 MS (Java/Others)    Memory Limit: 65536/32768 K (Java/Others)Total Submission(s): 6811    Accepted Submission(s): 3690Problem DescriptionAs we all know the Train Problem I, the boss of the Ignatius Train Station want to know if all the trains come in strict-increasing order, how many orders that all the trains can get out of the railway.InputThe input contains several test cases. Each test cases consists of a number N(1<=N<=100). The input is terminated by the end of file.OutputFor each test case, you should output how many ways that all the trains can get out of the railway.Sample Input12310Sample Output12516796HintThe result will be very large, so you may not process it by 32-bit integers. AuthorIgnatius.L*/#include<stdio.h>#include<string.h> int h[101][121];int main(){    int i,j,k,l,n;    memset(h, 0, sizeof(h));    h[0][1]=1;h[1][1]=1;    for(i = 2; i <= 100; ++i)//求第i个卡特兰数     {        for(j = 0; j < i; ++j)//断点         //大数乘法             for(k = 1; k < 60; ++k)//第j个的第k位 * 第i-j-1的每一位             for(l = 1; l < 60; ++l)        h[i][k + l - 1] += h[j][k] * h[i - j - 1][l];        for(j = 1; j < 60; ++j)//进位         {            h[i][j + 1] += h[i][j] / 10;            h[i][j] %= 10;        }    }    while(scanf("%d", &n) != EOF)    {        i=60;while( h[n][--i] == 0 );         for(j = i; j > 0; --j)//输出 printf("%d", h[n][j]);        printf("\n");    }    return 0;}

题意：火车按照严格递增的序列进站，求有多少种出战的顺序。

思路：假设第k节火车最后出站，那么比k小的数必须先出站，然后k进站，剩下的也必须在k之前出站，相当于说对于某个n，f【n】 = f【n-k-1】 * f【k-1】，而k由1~n进行遍历。由于f【n】较大，所以要用大数相乘，开一个二位数组，第一位保存n第二位保存答案。

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