Creating combinations based on possible pairs

We represent possible combinations by lists of length n. The indices for which we haven't got any information yet will contain None.

Each turn will treat a pairs of indices, and check all the rules for this pair.

If the first value of the pair exists in a possible combination of the previous turn, and the second one has never been touched (so is None), we add this as a new possible combination for this turn.

If both values exist in a previous combination, this confirms that it may be valid, we add it as well.

We can just abandon the results of the previous turn, as combinations that we previously considered as possible and haven't been validated on this turn are impossible.

So, the code:

from itertools import combinations

def possible_combs(n, pairs_list):
    # Pairs of indices, generated in the same order as the lists of allowed pairs
    indices = combinations(range(n), r=2)
    # Current list of possible combinations. None means no information for this index
    current = [[None] * n]

    for (first, last), allowed in zip(indices, pairs_list):
        previous = current
        current = []
        # Iteration on each allowed pair for the current pair of indices
        for i, j in allowed:
            for comb in previous:
                if comb[first] is None:
                    # We can have previous combinations having None for the starting index 
                    # only during the first step. In this case, we create the path. 
                    new = comb[:]
                    new[first] = i
                    new[last] = j
                    current.append(new)
                if comb[first] == i:
                    if comb[last] is None:
                        # A path leading to a yet unknown value, we add it
                        new = comb[:]
                        new[last] = j
                        current.append(new)
                    elif comb[last] == j:
                        # A valid path, we keep it
                        current.append(comb[:])
                    # At this point, any previous combination that didn't satisfy 
                    # any rule of this turn hasn't made it
                    # to current and will be forgotten...
    return current

Sample run on your data:

possible_combs(3, [[(1,2), (1,3), (2,2)],
                    [(2,2), (1,2), (2,4)],
                    [(1,2), (4,3), (2,2)]])

Output:

[[2, 2, 2], [1, 2, 2]]

Note that it makes no hypothesis on the number of rules for each pair of indices.

The problem can be depicted as a graph, where nodes are organised in columns:

A node is uniquely identified by the column it is in and the value it has.

A solution can be constructed by taking the possible edges between the first two columns, and then extending those edges to paths of size 2 where possible, involving the nodes of the relevant column, then extending again to size 3, involving the next column, ...etc. Each time a node is added to a path, it must be verified that all previous nodes in the path connect to that new node.

To do this efficiently I would suggest using an adjacency list type of data structure, or in fact an adjacency set, so that you can quickly fetch which nodes are reachable in a given column from a given node in another column. These sets of neighbors can be intersected so to be left with connections that fulfil all constraints.

I would define the input constraints as a dictionary, so there is no doubt as to what i and j are (the columns) for a given list of pairs. So the example input would be this dictionary:

{
    (0, 1): [(1,2), (1,3), (2,2)],
    (0, 2): [(2,2), (1,2), (2,4)],
    (1, 2): [(1,2), (4,3), (2,2)]
}

Code:

from collections import defaultdict

def solve(constraints):
    # n is the size of each output tuple 
    n = max(b for _, b in constraints) + 1

    # convert contraints to adjacency sets
    graph = {}
    for key, pairs in constraints.items():
        dct = defaultdict(set)
        for a, b in pairs:
            dct[a].add(b)
        graph[key] = dct

    paths = constraints[(0, 1)]
    for j in range(2, n):
        newpaths = []
        for path in paths:
            additions = graph[(0, j)][path[0]]
            for i in range(1, len(path)):
                additions &= graph[(i, j)][path[i]]
                if not additions:  # quick exit
                    break
            newpaths.extend((*path, num) for num in additions)
        paths = newpaths

    return paths

Call like this:

constraints = {
    (0, 1): [(1,2), (1,3), (2,2)],
    (0, 2): [(2,2), (1,2), (2,4)],
    (1, 2): [(1,2), (4,3), (2,2)]
}
result = solve(constraints)
print(result)

Output:

[(1, 2, 2), (2, 2, 2)]

You can use recursion while tracking the previous inclusion of values associated with the tuple pairs in your input. This way, the combinations can be more efficiently produced up-front, without having to filter an entire set of combinations afterwards:

from collections import defaultdict
d, d1 = {(0, 1): [(1, 2), (1, 3), (2, 2)], (0, 2): [(2, 2), (1, 2), (2, 4)], (1, 2): [(1, 2), (4, 3), (2, 2)]}, defaultdict(list)
for (a, b), j in d.items():
   d1[a].extend((l:=list(zip(*j)))[0])
   d1[b].extend(l[1])

def combos(p, c = {}):
   if not p:
      yield [*c.values()]
   else:
      for i in set(d1[p[0]]):
         if len(c) < 1 or all((b, i) in d[(a, p[0])] for a, b in c.items()):
            yield from combos(p[1:], {**c, p[0]:i})

print(list(combos([*{i for k in d for i in k}])))

Output:

[[1, 2, 2], [2, 2, 2]]