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Count K-Subsequences of a String With Maximum Beauty

Difficulty: Hard

Problem Description

You are given a string s and an integer k. A k-subsequence is a subsequence of s, having length k, and all its characters are unique. Let f(c) denote the number of times the character c occurs in s. The beauty of a k-subsequence is the sum of f(c) for every character c in the k-subsequence. The task is to return the number of k-subsequences whose beauty is the maximum among all k-subsequences. Since the answer may be too large, return it modulo 10^9 + 7.

Key Insights

  • A k-subsequence must consist of unique characters.
  • The beauty of the k-subsequence is determined by the frequency of its characters in the original string.
  • The maximum beauty can be derived from the top k most frequent characters.
  • Combinatorial counting is necessary to determine how many ways we can form these k-subsequences.

Space and Time Complexity

Time Complexity: O(n + k log k) where n is the length of the string (for counting frequencies and sorting). Space Complexity: O(1) for the frequency array since there are only 26 lowercase letters.

Solution

To solve this problem, we will follow these steps:

  1. Count the frequency of each character in the string s using a hash table (or array).
  2. Extract the frequencies and sort them in descending order.
  3. Select the top k frequencies to maximize the beauty.
  4. Calculate the maximum beauty as the sum of these top k frequencies.
  5. Count the number of ways to choose characters to form k-subsequences using combinatorial counting (n choose k).
  6. Return the result modulo 10^9 + 7.

We utilize a list to maintain the counts of characters and use sorting to efficiently find the top k frequencies.

Code Solutions

def countKSubsequencesWithMaxBeauty(s: str, k: int) -> int:
    from collections import Counter
    from math import comb

    MOD = 10**9 + 7
    freq = Counter(s)
    
    # Get the frequencies and sort them in descending order
    freq_values = sorted(freq.values(), reverse=True)
    
    # If k is greater than the number of unique characters, return 0
    if k > len(freq_values):
        return 0
    
    # Calculate the maximum beauty
    max_beauty = sum(freq_values[i] for i in range(k))
    
    # Count ways to choose k characters with the highest frequency
    ways = 1
    for i in range(k):
        count = freq_values[i]
        # Count how many times this count can be chosen from all occurrences
        ways *= count
        ways %= MOD
    
    return ways
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