More efficient methods for finding subsets of Subsets()

edit

Here is an iterator over 4-tuples (a, b, c, d) such that a + b + c + d = k and a < b < c < d. You can also ensure that d <= nmax, which seems a requirement of your problem.

def get_series(k, nmax=None):
    """
    Iterator over all series (a, b, c, d) such that:
    - a + b + c + d = k
    - a < b < c < d

    When nmax is specified, we furthermore ensure that a,b,c,d <= nmax.
    """
    if nmax is None:
        for a in range(1, (k - 6) // 4 + 1):
            for b in range(a + 1, (k - a - 3) // 3 + 1):
                for c in range(b + 1, (k - a - b - 1) // 2 + 1):
                    yield (a, b, c, k - a - b - c)
    else:
        for a in range(1, min(nmax, (k - 6) // 4) + 1):
            for b in range(a + 1, min(nmax, (k - a - 3) // 3) + 1):
                for c in range(b + 1, min(nmax, (k - a - b - 1) // 2) + 1):
                    d = k - a - b - c
                    if d <= nmax:
                        yield (a, b, c, d)

you can check that

This is way faster than your brute force attempt

sage: %time NList = [i for i in Subsets([1..100], 4) if sum(i) == 100]
CPU times: user 31 s, sys: 83.3 ms, total: 31.1 s
Wall time: 31.1 s
sage: %time L = list(get_series(100))
CPU times: user 7.23 ms, sys: 136 µs, total: 7.36 ms
Wall time: 7.35 ms
sage: len(L) == len(NList)
True
sage: L == [tuple(sorted(i)) for i in NList]
True

This is of course not sufficient for solving efficiently your problem since you have plenty of 4-tuples to consider and then triples of 4-tuples...

sage: %time L = list(get_series(1000))
CPU times: user 2.82 s, sys: 99.1 ms, total: 2.92 s
Wall time: 2.92 s
sage: len(L)
6840777

Another useful idea is to store the tuples in a dictionary with key a^2 + b^2 + c^2 + d^2 and value the list of corresponding tuples (so with same sum and same sum of squares).