1641. Count Sorted Vowel Strings

Question

Given an integer n, return the number of strings of length n that consist only of vowels (a, e, i, o, u) and are lexicographically sorted.

A string s is lexicographically sorted if for all valid i, s[i] is the same as or comes before s[i+1] in the alphabet.

Example 1:

Input: n = 1
Output: 5
Explanation: The 5 sorted strings that consist of vowels only are ["a","e","i","o","u"].

Example 2:

Input: n = 2
Output: 15
Explanation: The 15 sorted strings that consist of vowels only are
["aa","ae","ai","ao","au","ee","ei","eo","eu","ii","io","iu","oo","ou","uu"].
Note that "ea" is not a valid string since 'e' comes after 'a' in the alphabet.

Example 3:

Input: n = 33
Output: 66045

Constraints:

• 1 <= n <= 50

Summary

Find the pattern:

• When n=1:
  • “a”, “e”, “i”, “o”, “u”;
• When n=2:
  • “u” can be added before n=1 “u” start string (“uu”);
  • “o” can be added before n=1 “o”, “u” start string (“oo”, “ou”);
  • “i” can be added before n=1 “i”, “o”, “u” start string (“ii”, “io”, “iu”);
  • “e” can be added before n=1 “e”, “i”, “o”, “u” start string (“ee”, “ei”, “eo”, “eu”)
  • “a” can be added before n=1 “a”, “e”, “i”, “o”, “u” start string (“aa”, “ae”, “ai”, “ao”, “au”)
• When n=3:
  • “u” can be added before n=2 “u” start string (“uuu”);
  • “o” can be added before n=2 “o”, “u” start string (“ooo”, “oou”, “ouu”)
  • “i” can be added before n=2 “i”, “o”, “u” start string (“iii”, “iio”, “iiu”, “ioo”, “iou”, “iuu”)
  • “e” can be added before n=2 “e”, “i”, “o”, “u” start string (“eee”, “eei”, “eeo”, “eeu”, “eii”, “eio”, “eiu”, “eoo”, “eou”, “euu”)
  • “a” can be added before n=2 “a”, “e”, “i”, “o”, “u” start string (“aaa”, “aae”, “aai”, “aao”, “aau”, “aee”, “aei”, “aeo”, “aeu”, “aii”, “aio”, “aiu”, “aoo”, “aou”, “auu”)

…

• When n=N:
  • “u” can be added before n=N-1 “u” start string;
    • total(n=N, u) = total(n=N-1, u);
  • “o” can be added before n=N-1 “o”, “u” start string;
    • total(n=N, o) = total(n=N-1, o) + total(n=N-1, u);
  • “i” can be added before n=N-1 “i”, “o”, “u” start string;
    • total(n=N, i) = total(n=N-1, i) + total(n=N-1, o) + total(n=N-1, u);
  • “e” can be added before n=N-1 “e”, “i”, “o”, “u” start string;
    • total(n=N, e) = total(n=N-1, e) + total(n=N-1, i) + total(n=N-1, o) + total(n=N-1, u);
  • “a” can be added before n=N-1 “a”, “e”, “i”, “o”, “u” start string;
    • total(n=N, a) = total(n=N-1, a) + total(n=N-1, e) + total(n=N-1, i) + total(n=N-1, o) + total(n=N-1, u);

Solution

class Solution {
public:
    int countVowelStrings(int n) {
        int a=1, e=1, i=1, o=1, u=1;
        while (--n) {
            u = u;
            o += u;
            i += o;
            e += i;
            a += e;
        }
        return a+e+i+o+u;
    }
};