(6%) This problem considers permutations of $A_n = \{0,1,\ldots,n-1\}$. Let $P$ be a permutation from $A_n$. $P$ can be represented as a 0-1 string $S$ of length $n-1$, where:

• $S[i] = '1'$ if $P[i] < P[i+1]$, and
• $S[i] = '0'$ if $P[i] > P[i+1]$.

For example, a permutation $P = (0, 1, 3, 2)$ can be represented as a string 110. Note that this representation is not one to one, i.e., different permutations may be represented as the same string. Given a 0-1 string $S$, we are interested in reconstructing a possible permutation $P$.

Consider the following C++ procedure.

vector<int> findPermutation(string S) {
        int left = count(S.begin(), S.end(), '0'), right = left;
        vector<int> res = {left};
        for (char &c : S)
                res.push_back(c == '1' ? ++right : --left);
        return res;
}

Which of the following statements is/are correct?