trying to write a recursive function that counts the number of sequences that sum up to that number C++

Question

Okay, so here is what I\'m trying to do. The user inputs a number. I\'m trying to write a recursive function that counts the number of sequences that sum up to that number (

Answer 1

Define P(n) as the number of ways to partition n, restricting n to be an integer, n >= 1.

Define p(n, k) as the number of ways to partition n using numbers no larger than k, restricting k to be an integer, k >= 1, k <= n.

It follows that P(n) = sum (i: 1..n) p(n, i).

To find p(n, k), first note that we neatly avoid double-counting by simply keeping the partition sorted: take the largest chunk first. So the first chunk may have any size from 1 up to k, and then the rest of the chunks must account for the rest of the total n, and be no larger than the first chunk.

Thus p(n, k) = sum (i: 1..k) p(n - i, i).

As a base case, p(1, 1) = 1.

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A sample implementation, very much not guaranteed to be at all efficient, or even to compile (but it should give you the basic idea) - spoilers!

// A utility function to represent the result of appending to a vector,
// as a new vector (without affecting the previous one).
template 
vector operator<<(vector copy, T to_append) {
    // We passed by value to get a copy implicitly.
    copy.push_back(to_append);
    return copy;
}

// A utility function to append one vector to another.
template 
vector& operator+=(vector& original, const vector& to_append) {
    // 'copy' is in namespace std:: and defined in .
    // 'back_inserter' is in namespace std:: and defined in .
    copy(to_append.begin(), to_append.end(), back_inserter(original));
    return original;
}

vector > partitions(int remaining, int limit, vector prefix) {
    // Finds the partitions of some larger number which start with the
    // numbers in 'prefix', such that the rest of the "chunks" sum to
    // 'remaining' and are all no larger than 'limit'.

    // 'max' is in namespace std:: and defined in . We restrict
    // the 'limit' to be no more than 'remaining'.
    limit = max(remaining, limit);

    vector > result;

    // Base case. 
    if (remaining == 1) {
        return result << (prefix << 1); // note the parentheses are required!
    }

    for (int i = limit; i > 0; --i) {
        result += partitions(remaining - i, i, prefix << i);
    }

    return result;
}