Solution: Count the Number of Good Subsequences

Statement

Count and return the number of good subsequences in the given string s. You may return the modulo $10^9 + 7$ of the count.

Constraints:

• $1 \leq$s.length $\leq 10^4$

• s will only contain lowercase English characters.

Solution

The solution to this problem involves calculating the combinations. We will start by calculating the frequency of each character and then iterate over the frequencies, from 1 to the highest frequency of any character. We will count subsequences where each character appears for some or all of its frequency, i.e., calculating combinations. The count is summed up in each iteration, and after the loop terminates, 1 is deducted from the final count. The reason for deducting 1 is that the empty subsequence will also be counted. Because empty subsequence does not qualify as a good subsequence, we will remove its count from the final calculation.

Computing the combinations is a typical Binomial Coefficient problemThe binomial coefficient, often denoted as "n choose k," calculates the number of ways to choose k elements from a set of n distinct elements., known as n choose k problem, which is in mathematics written as $\binom{n}{k} = \frac{n!}{k! \cdot (n - k)!}$. Counting the combinations would require us to calculate the factorial and division by factorial, calculated by multiplying with the modular inverses.

Because calculating the factorial[i] requires calculating the [factorial[i-1]..factorial[2]] again and again, therefore we will use dynamic programming to calculate the factorial[i] by multiplying i only with factorial[i — 1]. That is, we will reuse the existing results instead of calculating them again.

The algorithm to solve this problem is as follows:

• Initialize two arrays: factorials and inverses of size n + 1, an integer array, character_frequencies, of the same size to count the frequencies of each character and an integer variable finalCount with $0$.

• Start a loop from $1$ to n + 1. Inside the loop, compute the factorial and inverses of each ith number and store it in factorials[i] and inverses[i], respectively.

• Start a loop from $0$ to n + 1 and count the frequencies of each character of s. Moreover, keep the record of maximum frequency and save it in max_frequency.

• Start a nested loop, where the outer loop, controlled by i, runs for max_frequency times and the inner loop, controlled by j runs for $26$ times.

  • Inside the outer loop, initialize a variable count with the value. Perform the following steps Inside the inner loop:

    • Compute the number of combinations, that is, n choose k, where n is the value of character_frequency[j], and k is the value of i. The number of combinations can be calculated with (factorials[n] * inverses[k] % MOD) * inverses[n-k] % MOD.

    • Add $1$ to the number of combinations. This is because we consider the possibility of not choosing a character when constructing subsequences. In other words, for each character that can be included in a subsequence, there are two choices: either include the character or do not. Adding $1$ captures these two possibilities. For example, if we have the string $aabbc$, we can either choose one $a$ and one $b$ or one $a$ and no $b$, or no $a$ and one $b$, or neither $a$ nor $b$. If we don't add $1$ to our calculation, we would miss the possibility of having a non-empty subsequence that excludes both $a$ and $b$.

    • Update the count by multiplying this calculation with the current value of the count.

  • Deduct $1$ from the count, add it to the current value of final_count, and take the modulus of this value. Update the final_count value with this new value. The reason to deduct $1$ from the count is that we need to exclude the empty subsequence.

• After the loop terminates, the value of final_count holds the number of good subsequences in the input string s.

Let’s look at the following illustration to get a better understanding of the solution: