Problem:
Score : 600 points

Problem Statement

You are given an integer sequence of length n+1, a_1,a_2,...,a_{n+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 10^9+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 \leq n \leq 10^5

1 \leq a_i \leq n

Each of the integers 1,...,n appears in the sequence.

n and a_i are integers.

Input

Input is given from Standard Input in the following format:

n
a_1 a_2 ... a_{n+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 10^9+7.

Sample Input 1

3
1 2 1 3

Sample Output 1

3
5
4
1

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.

Sample Input 2

1
1 1

Sample Output 2

1
1

There is one subsequence with length 1: 1.

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

Sample Input 3

32
29 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

32
525
5453
40919
237336
1107568
4272048
13884156
38567100
92561040
193536720
354817320
573166440
818809200
37158313
166803103
166803103
37158313
818809200
573166440
354817320
193536720
92561040
38567100
13884156
4272048
1107568
237336
40920
5456
528
33
1

Be sure to print the numbers modulo 10^9+7.

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

#pragma warning disable

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Runtime.Serialization.Formatters.Binary;
using System.Text;
using System.Text.RegularExpressions;
using System.Diagnostics;
using System.Numerics;
using System.Collections;

static class MainClass
{
    public static void Main(string[] args)
    {
        SetFactor(100100);
        var n = Console.ReadLine().ToInt32();
        var an = Console.ReadLine().SplittedStringToInt32List();
        var andic = an.ToCountedDictionary();
        var esp = andic.Where(x => x.Value == 2).First();
        var esp1ind = an.FindIndex(x => x == esp.Key);
        var esp2ind = an.FindLastIndex(x => x == esp.Key);
        for (int i = 1; i <= n + 1; i++)
        {
            if (i == 1)
            {
                Console.WriteLine(n);
            }
            else if (i == n + 1)
            {
                Console.WriteLine(1);
            }
            else
            {
                var sum = 0;
                var num1 = GetCombination(n + 1, i);
                var newi = i - 1;
                for (int j = 0; j <= newi; j++)
                {
                    if (esp1ind < j || n - esp2ind < newi - j)
                        continue;
                    var num2 = GetCombination(esp1ind, j);
                    var num3 = GetCombination(n - esp2ind, newi - j);
                    var num4 = (- 1 * num2 * num3) % Mod1e9;
                    num1 += num4;
                }
                Console.WriteLine(num1);
            }
        }

        Console.ReadLine();
        //var nm = Console.ReadLine().SplittedStringToInt32List();
        //var n = nm[0];
        //var m = nm[1];
        //InitializeUnionFind(n);
        //var abs = new List<KeyValuePair<int,int>>();
        //for (int i = 0; i < m; i++)
        //{
        //    var ab = Console.ReadLine().SplittedStringToInt32List();
        //    abs.Add(new KeyValuePair<int, int>(ab[0] - 1, ab[1] - 1));
        //}
        //abs.Reverse();
        //var ls = new List<int>();
        //foreach (var item in abs)
        //{
        //    Unite(item.Key, item.Value);
        //    ls.Add(ParentCount);
        //}
        //ls.Reverse();
        //var before = -1;
        //var befores = 0;
        //for (int i = 0; i < m; i++)
        //{
        //    if (before != -1)
        //    {
        //        var num = ls[i] - befores;
        //        Console.WriteLine((num * (num - 1))/2 + before);
        //        before = (num * (num - 1)) / 2 + before;
        //    }
        //    else
        //    {
        //        var num = ls[i] - befores;
        //        Console.WriteLine((num * ( num - 1)) / 2);
        //        before = (num * (num - 1)) / 2;
        //    }
        //    befores = ls[i];
        //}

        //Console.ReadLine();
    }


    public static BigInteger[] factors;
    public static BigInteger[] revFactors;
    public static void SetFactor(int N)
    {
        factors = new BigInteger[N];
        factors[0] = 1;
        revFactors = new BigInteger[N];
        for (int i = 1; i < N; i++)
        {
            factors[i] = factors[i - 1] * i;
            factors[i] %= Mod1e9;
        }
        revFactors[N - 1] = BigInteger.ModPow(factors[N - 1], Mod1e9 - 2, Mod1e9);
        for (int i = N - 2; i >= 0; i--)
        {
            revFactors[i] = revFactors[i + 1] * (i + 1);
            revFactors[i] %= Mod1e9;
        }
    }
    public static BigInteger GetCombination(int a, int b)
    {
        return (((factors[a] * revFactors[b]) % Mod1e9) * revFactors[a - b]) % Mod1e9;
    }

    public static int ParentCount;

    public static int[] Parents;
    public static int[] IfParentsChildrenCount;

    public static int[] Ranks;

    public static void InitializeUnionFind(int N)
    {
        Parents = new int[N];
        Ranks = new int[N];
        IfParentsChildrenCount = new int[N];
        for (int i = 0; i < N; i++)
        {
            Parents[i] = i;
            Ranks[i] = 0;
            IfParentsChildrenCount[i] = 1;
        }
        ParentCount = N + 1;
    }

    public static int Root(int a)
    {
        if (Parents[a] == a)
        {
            return a;
        }
        else
        {
            Parents[a] = Root(Parents[a]);
            return Parents[a];
        }
    }

    public static bool Same(int a, int b)
    {
        return Root(a) == Root(b);
    }

    public static void Unite(int a, int b)
    {
        var parentA = Root(a);
        var parentB = Root(b);
        if (parentA == parentB)
            return;
        if (Ranks[parentA] < Ranks[parentB])
        {
            Parents[parentA] = parentB;
            IfParentsChildrenCount[parentB] += IfParentsChildrenCount[parentA];
            ParentCount -= IfParentsChildrenCount[parentA];

            IfParentsChildrenCount[parentA] = 0;
        }
        else
        {
            Parents[parentB] = parentA;
            if (Ranks[parentA] == Ranks[parentB])
            {
                Ranks[parentA]++;
            }
            IfParentsChildrenCount[parentA] += IfParentsChildrenCount[parentB];
            ParentCount -= IfParentsChildrenCount[parentB];
            IfParentsChildrenCount[parentB] = 0;
        }
    }

    #region ライブラリ
    public static long ToInt64(this string str) => long.Parse(str);
    public static int ToInt32(this string str) => int.Parse(str);
    public static BigInteger ToBigInteger(this string str) => BigInteger.Parse(str);
    public static List<string> SplittedStringToList(this string str) => str.Split(new[] { ' ' }, StringSplitOptions.RemoveEmptyEntries).ToList();
    public static List<int> SplittedStringToInt32List(this string str) => str.Split(new[] { ' ' }, StringSplitOptions.RemoveEmptyEntries).Select(x => int.Parse(x)).ToList();
    public static List<long> SplittedStringToInt64List(this string str) => str.Split(new[] { ' ' }, StringSplitOptions.RemoveEmptyEntries).Select(x => long.Parse(x)).ToList();
    public static List<BigInteger> SplittedStringToBigInteger(this string str) => str.Split(new[] { ' ' }, StringSplitOptions.RemoveEmptyEntries).Select(x => BigInteger.Parse(x)).ToList();
    public const int INF = int.MaxValue / 4;
    public const long LONGINF = long.MaxValue / 2;
    public const int Mod1e9 = 1000000007;

    public static void PrintArray(bool[,] array)
    {
        for (int i = 0; i < array.GetLength(0); i++)
        {
            var sb = new StringBuilder();
            for (int j = 0; j < array.GetLength(1); j++)
            {
                sb.Append(array[i, j]).Append(" ");
            }
            Console.WriteLine(sb.ToString());
        }
    }
    public static int Manhattan(int x1, int x2, int y1, int y2)
    {
        return Math.Abs(x1 - x2) + Math.Abs(y1 - y2);
    }

    public static Dictionary<T, int> ToCountedDictionary<T>(this IEnumerable<T> items)
    {
        var dic = new Dictionary<T, int>();
        foreach (var item in items)
        {
            if (dic.ContainsKey(item))
                dic[item]++;
            else
                dic.Add(item, 1);
        }
        return dic;
    }

    public static long Sums(IEnumerable<int> list)
    {
        var sum = 0l;
        foreach (var item in list)
        {
            sum += item;
        }
        return sum;
        #endregion
    }
}