Sergio is analyzing the merge algorithm for his algorithms assignment. His professor has written the following algorithm:

std::vector<int> merge(std::vector<int> A, std::vector<int> B) {
    std::vector<int> C;
    int i=0, j=0;
    while (i < A.size() && j < B.size()) {
        if (A[i] <= B[j]) {
            C.push_back(A[i]);
            i += 1;
        } else {
            C.push_back(B[j]);
            j += 1;
        }
    }
    while (i < A.size()) {
        C.push_back(A[i]);
        i += 1;
    }
    while (j < B.size()) {
        C.push_back(B[j]);
        j += 1;
    }   
    return C;
}

To do this, he has $$$M$$$ non-empty sequences $$$S_1$$$, $$$S_2$$$, ..., $$$S_M$$$, and performs the following operation $$$S_1 =$$$ merge($$$S_1, S_i$$$) for $$$i$$$ from $$$2$$$ to $$$M$$$. Thus, finally, to carry out his analysis, he wants to count the number of distinct valid initial sequences that produce the same final sequence $$$S_1$$$.

Two initial sequences (sequences of sequences) $$$S_1, S_2, \ldots, S_M$$$ and $$$S'_1, S'_2, \ldots, S'_m$$$ are distinct if there exists an index $$$i$$$ ($$$1 \le i \le M$$$) such that $$$S_i \neq S'_i$$$. Similarly, two sequences $$$A$$$ and $$$B$$$ are distinct if they differ in their number of elements or there exists an index $$$j$$$ ($$$1 \le j \le |A_i|$$$) such that $$$A_j \neq B_j$$$.

Note that the final $$$S_1$$$ sequence is not necessarily sorted.