Problem:
Score : 1100 points

Problem Statement

You are given integers N, K, and an integer sequence A of length M.

An integer sequence where each element is between 1 and K (inclusive) is said to be colorful when there exists a contiguous subsequence of length K of the sequence that contains one occurrence of each integer between 1 and K (inclusive).

For every colorful integer sequence of length N, count the number of the contiguous subsequences of that sequence which coincide with A, then find the sum of all the counts.
Here, the answer can be extremely large, so find the sum modulo 10^9+7.

Constraints

1 \leq N \leq 25000

1 \leq K \leq 400

1 \leq M \leq N

1 \leq A_i \leq K

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K M
A_1 A_2 ... A_M

Output

For every colorful integer sequence of length N, count the number of the contiguous subsequences of that sequence which coincide with A, then print the sum of all the counts modulo 10^9+7.

Sample Input 1

3 2 1
1

Sample Output 1

9

There are six colorful sequences of length 3: (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2) and (2,2,1).
The numbers of the contiguous subsequences of these sequences that coincide with A=(1) are 2, 2, 1, 2, 1 and 1, respectively.
Thus, the answer is their sum, 9.

Sample Input 2

4 2 2
1 2

Sample Output 2

12

Sample Input 3

7 4 5
1 2 3 1 2

Sample Output 3

17

Sample Input 4

5 4 3
1 1 1

Sample Output 4

0

Sample Input 5

10 3 5
1 1 2 3 3

Sample Output 5

1458

Sample Input 6

25000 400 4
3 7 31 127

Sample Output 6

923966268

Sample Input 7

9954 310 12
267 193 278 294 6 63 86 166 157 193 168 43

Sample Output 7

979180369

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Text;
using System.Collections.Generic;
using System.Linq;
class Solve{
    public Solve(){}
    StringBuilder sb;
    ReadData re;
    public static int Main(){
        new Solve().Run();
        return 0;
    }
    void Run(){
        sb = new StringBuilder();
        re = new ReadData();
        Calc();
        Console.Write(sb.ToString());
    }
    void Calc(){
        int N = re.i();
        int K = re.i();
        int M = re.i();
        int[] A = re.ia();
        if(N < K){
            sb.Append("0\n");
            return;
        }
        for(int i=0;i<M;i++){
            A[i]--;
        }
        Fact F = new Fact(N);
        long count = N-M+1;
        for(int i=0;i<N-M;i++){
            count *= K;
            count %= Define.mod;
        }
        int left = M+1;
        {
            bool[] used = new bool[K];
            for(int i=0;i<M;i++){
                if(used[A[i]]){
                    left = i;
                    break;
                }
                used[A[i]] = true;
            }
        }
        int right = -1;
        {
            bool[] used = new bool[K];
            for(int i=M-1;i>=0;i--){
                if(used[A[i]]){
                    right = i;
                    break;
                }
                used[A[i]] = true;
            }
        }
        int mc = MaxColor(A,K);
        if(M >= K){
            if(mc == K){
                sb.Append(count+"\n");
                return;
            }   
        }
        else if(mc == M){
            long[,] DP = new long[N+1,K];
            long[,] DP2 = new long[N+1,K];
            DP[0,0] = 1;
            for(int i=1;i<=N;i++){
                long sum = 0;
                long sum2 = 0;
                for(int j=K-1;j>=1;j--){
                    sum += DP[i-1,j];
                    sum %= Define.mod;
                    sum2 += DP2[i-1,j];
                    sum2 %= Define.mod;
                    DP[i,j] = (DP[i-1,j-1]*(K-j+1) + sum) % Define.mod;
                    DP2[i,j] = (DP2[i-1,j-1]*(K-j+1) + sum2) % Define.mod;
                    if(j >= M){
                        DP2[i,j] = (DP2[i,j] + DP[i,j]) % Define.mod;
                    }
                }
            }
            long c = 0;
            for(int i=0;i<K;i++){
                c += DP2[N,i];
            }
            c %= Define.mod;
            count = (count + Define.mod - (c * F.rf[K] % Define.mod * F.f[K-M] % Define.mod)) % Define.mod;
            sb.Append(count+"\n");
            return;
        }
        long[,] DPL = new long[N-M+1,K];
        DPL[0,left] = 1;
        for(int i=1;i<=N-M;i++){
            long sum = 0;
            for(int j=K-1;j>=1;j--){
                sum += DPL[i-1,j];
                sum %= Define.mod;
                DPL[i,j] = (DPL[i-1,j-1]*(K-j+1) + sum) % Define.mod;
            }
            //DPL[i,j] = DPL[i-1,j-1]*(K-j+2) + sum k=j... DPL 
        }
        long[,] DPR = new long[N-M+1,K];
        DPR[0,M-1-right] = 1;
        for(int i=1;i<=N-M;i++){
            long sum = 0;
            for(int j=K-1;j>=1;j--){
                sum += DPR[i-1,j];
                sum %= Define.mod;
                DPR[i,j] = (DPR[i-1,j-1]*(K-j+1) + sum) % Define.mod;
            }
        }
        for(int i=0;i<=N-M;i++){
            long sum1 = 0;
            long sum2 = 0;
            for(int j=0;j<K;j++){
                sum1 += DPL[i,j];
                sum2 += DPR[N-M-i,j];
            }
            sum1 %= Define.mod;
            sum2 %= Define.mod;
            count = (count + Define.mod - sum1*sum2%Define.mod) % Define.mod;
        }
        sb.Append(count+"\n");
    }
    int MaxColor(int[] A,int K){
        int M = A.Length;
        int[] counter = new int[K];
        int max = 0;
        int kind = 0;
        for(int i=0;i<K && i<M;i++){
            if(counter[A[i]] == 0){
                kind++;
            }
            counter[A[i]]++;
            max = Math.Max(max,kind);
        }
        for(int i=K;i<M;i++){
            counter[A[i-K]]--;
            if(counter[A[i-K]] == 0){
                kind--;
            }
            if(counter[A[i]] == 0){
                kind++;
            }
            counter[A[i]]++;
            max = Math.Max(max,kind);
        }
        return max;
    }
}
class Fact{
    public long[] f;
    public long[] rf;
    public Fact(int N){
        f = new long[N+1];
        rf = new long[N+1];
        for(int i=0;i<N+1;i++){
            if(i == 0){
                f[i] = 1;
            }
            else{
                f[i] = (f[i-1]*i)%Define.mod;
            }
        }
        for(int i=N;i>=0;i--){
            if(i == N){
                rf[i] = pow(f[N],Define.mod-2);
            }
            else{
                rf[i] = rf[i+1]*(i+1)%Define.mod;
            }
        }
    }
    public long pow(long N,long K){
        if(K == 0){
            return 1;
        }
        else if(K % 2 == 0){
            long t = pow(N,K/2);
            return t*t%Define.mod;
        }
        else{
            return N*pow(N,K-1)%Define.mod;
        }
    }
    public long GetFact(int N){
        return f[N];
    }
    public long GetPerm(int N,int R){
        return f[N] * rf[N-R] % Define.mod;
    }
    public long GetConv(int N,int R){
        return ((f[N]*rf[R])%Define.mod*rf[N-R])%Define.mod;
    }
    public long GetRev(int N){
        if(N == 0){
            return 1;
        }
        else{
            return rf[N] * f[N-1] % Define.mod;
        }
    }
}
class ReadData{
    string[] str;
    int counter;
    public ReadData(){
        counter = 0;
    }
    public string s(){
        if(counter == 0){
            str = Console.ReadLine().Split(' ');
            counter = str.Length;
        }
        counter--;
        return str[str.Length-counter-1];
    }
    public int i(){
        return int.Parse(s());
    }
    public long l(){
        return long.Parse(s());
    }
    public double d(){
        return double.Parse(s());
    }
    public int[] ia(int N){
        int[] ans = new int[N];
        for(int j=0;j<N;j++){
            ans[j] = i();
        }
        return ans;
    }
    public int[] ia(){
        str = Console.ReadLine().Split(' ');
        counter = 0;
        int[] ans = new int[str.Length];
        for(int j=0;j<str.Length;j++){
            ans[j] = int.Parse(str[j]);
        }
        return ans;
    }
    public long[] la(int N){
        long[] ans = new long[N];
        for(int j=0;j<N;j++){
            ans[j] = l();
        }
        return ans;
    }
    public long[] la(){
        str = Console.ReadLine().Split(' ');
        counter = 0;
        long[] ans = new long[str.Length];
        for(int j=0;j<str.Length;j++){
            ans[j] = long.Parse(str[j]);
        }
        return ans;
    }
    public double[] da(int N){
        double[] ans = new double[N];
        for(int j=0;j<N;j++){
            ans[j] = d();
        }
        return ans;
    }
    public double[] da(){
        str = Console.ReadLine().Split(' ');
        counter = 0;
        double[] ans = new double[str.Length];
        for(int j=0;j<str.Length;j++){
            ans[j] = double.Parse(str[j]);
        }
        return ans;
    }
    public List<int>[] g(int N,int[] f,int[] t){
        List<int>[] ans = new List<int>[N];
        for(int j=0;j<N;j++){
            ans[j] = new List<int>();
        }
        for(int j=0;j<f.Length;j++){
            ans[f[j]].Add(t[j]);
            ans[t[j]].Add(f[j]);
        }
        return ans;
    }
    public List<int>[] g(int N,int M){
        List<int>[] ans = new List<int>[N];
        for(int j=0;j<N;j++){
            ans[j] = new List<int>();
        }
        for(int j=0;j<M;j++){
            int f = i()-1;
            int t = i()-1;
            ans[f].Add(t);
            ans[t].Add(f);
        }
        return ans;
    }
    public List<int>[] g(){
        int N = i();
        int M = i();
        List<int>[] ans = new List<int>[N];
        for(int j=0;j<N;j++){
            ans[j] = new List<int>();
        }
        for(int j=0;j<M;j++){
            int f = i()-1;
            int t = i()-1;
            ans[f].Add(t);
            ans[t].Add(f);
        }
        return ans;
    }
}
public static class Define{
    public const long mod = 1000000007;
}
public static class Debug{
    public static void Print(double[,,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                for(int l=0;l<k.GetLength(2);l++){
                    Console.Write(k[i,j,l]+" ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
        }
    }
    public static void Print(double[,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                Console.Write(k[i,j]+" ");
            }
            Console.WriteLine();
        }
    }
    public static void Print(double[] k){
        for(int i=0;i<k.Length;i++){
            Console.WriteLine(k[i]);
        }
    }
    public static void Print(long[,,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                for(int l=0;l<k.GetLength(2);l++){
                    Console.Write(k[i,j,l]+" ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
        }
    }
    public static void Print(long[,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                Console.Write(k[i,j]+" ");
            }
            Console.WriteLine();
        }
    }
    public static void Print(long[] k){
        for(int i=0;i<k.Length;i++){
            Console.WriteLine(k[i]);
        }
    }
    public static void Print(int[,,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                for(int l=0;l<k.GetLength(2);l++){
                    Console.Write(k[i,j,l]+" ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
        }
    }
    public static void Print(int[,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                Console.Write(k[i,j]+" ");
            }
            Console.WriteLine();
        }
    }
    public static void Print(int[] k){
        for(int i=0;i<k.Length;i++){
            Console.WriteLine(k[i]);
        }
    }
}