Problem explanation

We are given a string of n letters 'D' or 'I' and our job is to find a permutation for the numbers between 0 and n that matches with the string. The rules of the permutation is that a 'D' would mean the number before is greater than the one after and 'I' means the number before is less than the one after. An example can be that for the string "DID", the resulting permutations can be (1, 0, 3, 2), (2, 0, 3, 1), (2, 1, 3, 0), (3, 0, 2, 1), (3, 1, 2, 0), totaling 5. The idea is to return the amount of valid permutations the string can generate.

Solution explanation

Let dp[i][j] be the number of valid permutations with i + 1 digits, where the ith character is the jth largest unused digit. This is a 2D dynamic programming solution that builds up the number valid permutations bearing in mind the constraints of the problem.

If the ith character is 'I', then we sum the values of dp[i - 1][k] for all k >= j.

If the ith character is 'D', then we sum the values of dp[i - 1][k] for all k < j.

To speed up this process, we can calculate prefix and postfix sums of the dp array.

Algorithm Steps

The algorithm follows these main steps:

1. First we initialize our dp table. All row and column will be filled with zeros except for the first row as we are sure when we have one character we will always have a permutation.
2. We then build our dp table. In this table, for each character in the string we build accumulating prefix sums if the character is a 'D' or postfix (backward) sum for an 'I'. In order to do this we loop starting from i = 1 up to n. For each iteration of i, check if the i-1 character in the string is an 'I': 3. If the character is an 'I', then we are looking for increasing permutations. We start by initializing a postfixSum to zero and enter a loop from j = n - i down to zero. We then: add dp[i - 1][j + 1] to the postfix sum and then store the postfix sum in dp[i][j]. 4. If the character is 'D', then we are looking for decreasing permutations. Initialize a variable prefix to 0. Loop j from zero to n - i. Then: add dp[i - 1][j] to prefix and store prefix in dp[i][j].
3. Finally, return the first value of the last row in the dp table.

Python Solution

1
2python
3class Solution:
4    def numPermsDISequence(self, s: str) -> int:
5        MOD = 10**9 + 7
6        n = len(s)
7        dp = [[0] * (n + 1) for _ in range(n + 1)]
8        
9        # when there's only one digit, the number of permutations is 1
10        for j in range(n + 1):
11            dp[0][j] = 1
12        
13        for i in range(1, n + 1):
14            if s[i - 1] == 'I':
15                # Calculate postfix sum to prevent duplicate calculation
16                postfix_sum = 0
17                for j in range(n - i, -1, -1):
18                    postfix_sum = (postfix_sum + dp[i - 1][j + 1]) % MOD
19                    dp[i][j] = postfix_sum
20            else:
21                # Calculate prefix sum to prevent duplicate calculation
22                prefix_sum = 0
23                for j in range(n - i + 1):
24                    prefix_sum = (prefix_sum + dp[i - 1][j]) % MOD
25                    dp[i][j] = prefix_sum
26        return dp[n][0]

Java Solution

1
2java
3import java.util.Arrays;
4
5class Solution {
6    private static final int MOD = 1_000_000_007;
7    public int numPermsDISequence(String s) {
8        int n = s.length();
9        int[][] dp = new int[n + 1][n + 1];
10        
11        // when there's only one digit, the number of permutations is 1
12        Arrays.fill(dp[0], 1);
13        
14        for (int i = 1; i <= n; i++) {
15            if (s.charAt(i - 1) == 'I') {
16                int postfixSum = 0;
17                for (int j = n - i; j >= 0; j--) {
18                    postfixSum = (postfixSum + dp[i - 1][j + 1]) % MOD;
19                    dp[i][j] = postfixSum;
20                }
21            } else {
22                int prefixSum = 0;
23                for (int j = 0; j <= n - i; j++) {
24                    prefixSum = (prefixSum + dp[i - 1][j]) % MOD;
25                    dp[i][j] = prefixSum;
26                }
27            }
28        }
29        return dp[n][0];
30    }
31}

C++ Solution

1
2cpp
3class Solution {
4public:
5    int numPermsDISequence(string S) {
6        int n = S.size(), mod = 1e9+7;
7        vector<vector<int>> dp(n+1, vector<int> (n+1, 0));
8        
9        // when there's only one digit, the number of permutations is 1
10        for (int j = 0; j <= n; j++) dp[0][j] = 1;
11        
12        for (int i = 1; i <= n; i++)
13            if (S[i-1] == 'I') {
14                int tmp = 0;
15                for (int j = n - i ; j >= 0; --j) {
16                    tmp = (tmp + dp[i - 1][j + 1]) % mod;
17                    dp[i][j] = tmp;
18                }
19            } else {
20                int tmp = 0;
21                for (int j = 0; j <= n - i; ++j) {
22                    tmp = (tmp + dp[i - 1][j]) % mod;
23                    dp[i][j] = tmp;
24                }
25            }
26        return dp[n][0];
27    }
28};

C# Solution

1
2csharp
3public class Solution {
4    public int NumPermsDISequence(string s) {
5        int n = s.Length;
6        int mod = 1000000007;
7        
8        var dp = new int[n + 1, n + 1];
9        
10        // when there's only one digit, the number of permutations is 1
11        for (int j = 0; j <= n; j++)
12            dp[0, j] = 1;
13        
14        for (int i = 1; i <= n; i++)
15            if (s[i - 1] == 'I') {
16                int tmp = 0;
17                for (int j = n - i; j >= 0; --j) {
18                    tmp = (tmp + dp[i - 1, j + 1]) % mod;
19                    dp[i, j] = tmp;
20                }
21            } else {
22                int tmp = 0;
23                for (int j = 0; j <= n - i; ++j) {
24                    tmp = (tmp + dp[i - 1, j]) % mod;
25                    dp[i, j] = tmp;
26                }
27            }
28        
29        return dp[n, 0];
30    }
31}

JavaScript Solution

1
2javascript
3var numPermsDISequence = function(s) {
4    const MOD = 1e9 + 7;
5    const n = s.length;
6    let dp = Array.from(Array(n + 1), () => Array(n + 1).fill(0));
7    
8    // when there's only one digit, the number of permutations is 1
9    for (let j = 0; j <= n; j++)
10        dp[0][j] = 1;
11    
12    for(let i = 1; i <= n; i++)
13        if(s.charAt(i - 1) == 'I') {
14            let sum = 0;
15            for(let j = n - i; j >= 0; j--) {
16                sum = (sum + dp[i - 1][j + 1]) % MOD;
17                dp[i][j] = sum;
18            }
19        } else {
20            let sum = 0;
21            for(let j = 0; j <= n - i; j++) {
22                sum = (sum + dp[i - 1][j]) % MOD;
23                dp[i][j] = sum;
24            }
25        }
26    return dp[n][0];
27};

Applications & Conclusions

The problem involves arranging a permutation that fits a specific pattern represented by a string composed of 'D's and 'I's. It demonstrates a practical use of dynamic programming, a strategy often used in algorithms related to optimization or counting. Understanding the problem and solution can help improve your skill in handling array operations, dynamic programming, permutation explore, and the usage of prefix or postfix sum technique.

This problem can be applied in different problems such as scheduling or arranging tasks, ordering a list of data, and so on. Additionally, techniques used in this problem can be utilized in other more complex combinatory problems, such as those involving permutation and combinations, tasks scheduling, project sequencing, and more.

While the solution itself may not be directly implementable in many practical scenarios due to its specific nature, the process of finding the solution provides good practice in understanding the concept behind dynamic programming and postfix and prefix cumulative sums. The solution might seem tedious due to the deep nested loop, but with a good understanding of dynamic programming, arriving at the optimal solution would be much easier.

In data structures and algorithms, understanding how to break down problems into smaller subproblems that can be built up to arrive at the grand solution is a crucial skill. It also demonstrates how problems can be solved by considering previously solved subproblems rather than starting from scratch.

Overall, understanding this problem could be very beneficial, given you took time to understand not just the solution, but also the thought process behind arriving at the optimal solution. The implementation could be done in different programming languages including Python, Java, JavaScript and C#, promoting good practice for multilingual programmers.

While the problem is related to finding a permutation that satisfies a represented condition, the knowledge and skills gained from understanding the problem are applicable in other combinatorics problems and other disciplines.