AtCoder：11（数论 & 思维）

D - 11

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Time limit : 2sec / Memory limit : 256MB

Score : 600 points

Problem Statement

You are given an integer sequence of length n+1, a1,a2,…,an+1, which consists of the n integers 1,…,n. It is known that each of the n integers 1,…,n appears at least once in this sequence.

For each integer k=1,…,n+1, find the number of the different subsequences (not necessarily contiguous) of the given sequence with length k, modulo 109+7.

Notes

• If the contents of two subsequences are the same, they are not separately counted even if they originate from different positions in the original sequence.

• A subsequence of a sequence a with length k is a sequence obtained by selecting k of the elements of a and arranging them without changing their relative order. For example, the sequences 1,3,5 and 1,2,3 are subsequences of 1,2,3,4,5, while 3,1,2 and 1,10,100 are not.

Constraints

• 1≤n≤105
• 1≤ai≤n
• Each of the integers 1,…,n appears in the sequence.
• n and ai are integers.

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Input

Input is given from Standard Input in the following format:

na1 a2 ... an+1

Output

Print n+1 lines. The k-th line should contain the number of the different subsequences of the given sequence with length k, modulo 109+7.

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Sample Input 1

Copy

31 2 1 3

Sample Output 1

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3541

There are three subsequences with length 1: 1 and 2 and 3.

There are five subsequences with length 2: 1,1 and 1,2 and 1,3 and 2,1 and 2,3.

There are four subsequences with length 3: 1,1,3 and 1,2,1 and 1,2,3 and 2,1,3.

There is one subsequence with length 4: 1,2,1,3.

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Sample Input 2

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11 1

Sample Output 2

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11

There is one subsequence with length 1: 1.

There is one subsequence with length 2: 1,1.

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Sample Input 3

Copy

3229 19 7 10 26 32 27 4 11 20 2 8 16 23 5 14 6 12 17 22 18 30 28 24 15 1 25 3 13 21 19 31 9

Sample Output 3

Copy

3252554534091923733611075684272048138841563856710092561040193536720354817320573166440818809200371583131668031031668031033715831381880920057316644035481732019353672092561040385671001388415642720481107568237336409205456528331

Be sure to print the numbers modulo 109+7.

题意：给N+1个数，范围在[1,n]，且[1,n]每个数最少出现一次，问[1,n]长度的子序列各有多少种，结果模1e9+7。

思路：显然有一个数是重复出现的，那么需要处理下重复的。假如第i和i+3个数是一样的，此时计算子序列长度为x，在i前面和i+3后面选x-1个数就是重复的部分，减去它即可

，组合数计算用逆元。

# include <bits/stdc++.h>using namespace std;typedef long long LL;const LL mod = 1e9+7;const LL maxn = 1e5+3;LL inv[maxn+8]={1,1}, fi[maxn+8]={1,1}, fac[maxn+8]={1,1};int vis[maxn+8]={0};void init(){    for(int i=2; i<=maxn; ++i)    {        fac[i] = fac[i-1]*i%mod;        inv[i] = (mod-mod/i)*inv[mod%i]%mod;        fi[i] = fi[i-1]*inv[i]%mod;    }}LL c(LL n, LL m){    return fac[n]*fi[n-m]%mod*fi[m]%mod;}int main(){    init();    int n, t, dis;    scanf("%d",&n);    for(int i=1; i<=n+1; ++i)    {        scanf("%d",&t);        if(vis[t])            dis = vis[t]+n-i;        vis[t] = i;    }    for(int i=1; i<=n+1; ++i)    {        LL ans = c(n*1LL+1, i*1LL)%mod;        if(dis >= i-1) ans = (ans-c(dis*1LL, i*1LL-1)+mod)%mod;        printf("%lld\n",ans);    }    return 0;}