Here is a function I would like to write but am unable to do so. Even if you don't / can't give a solution I would be grateful for tips. For example, I know that there is a correlation between the ordered represantions of the sum of an integer and ordered set partitions but that alone does not help me in finding the solution. So here is the description of the function I need:

The Task

Create an efficient* function

List<int[]> createOrderedPartitions(int n_1, int n_2,..., int n_k)

that returns a list of arrays of all set partions of the set {0,...,n_1+n_2+...+n_k-1} in number of arguments blocks of size (in this order) n_1,n_2,...,n_k (e.g. n_1=2, n_2=1, n_3=1 -> ({0,1},{3},{2}),...).

Here is a usage example:

int[] partition = createOrderedPartitions(2,1,1).get(0);
partition[0]; // -> 0
partition[1]; // -> 1
partition[2]; // -> 3
partition[3]; // -> 2

Note that the number of elements in the list is (n_1+n_2+...+n_n choose n_1) * (n_2+n_3+...+n_n choose n_2) * ... * (n_k choose n_k). Also, createOrderedPartitions(1,1,1) would create the permutations of {0,1,2} and thus there would be 3! = 6 elements in the list.

* by efficient I mean that you should not initially create a bigger list like all partitions and then filter out results. You should do it directly.

Extra Requirements

If an argument is 0 treat it as if it was not there, e.g. createOrderedPartitions(2,0,1,1) should yield the same result as createOrderedPartitions(2,1,1). But at least one argument must not be 0. Of course all arguments must be >= 0.

Remarks

The provided pseudo code is quasi Java but the language of the solution doesn't matter. In fact, as long as the solution is fairly general and can be reproduced in other languages it is ideal.

Actually, even better would be a return type of List<Tuple<Set>> (e.g. when creating such a function in Python). However, then the arguments wich have a value of 0 must not be ignored. createOrderedPartitions(2,0,2) would then create

[({0,1},{},{2,3}),({0,2},{},{1,3}),({0,3},{},{1,2}),({1,2},{},{0,3}),...]

Background

I need this function to make my mastermind-variation bot more efficient and most of all the code more "beautiful". Take a look at the filterCandidates function in my source code. There are unnecessary / duplicate queries because I'm simply using permutations instead of specifically ordered partitions. Also, I'm just interested in how to write this function.

My ideas for (ugly) "solutions"

Create the powerset of {0,...,n_1+...+n_k}, filter out the subsets of size n_1, n_2 etc. and create the cartesian product of the n subsets. However this won't actually work because there would be duplicates, e.g. ({1,2},{1})...

First choose n_1 of x = {0,...,n_1+n_2+...+n_n-1} and put them in the first set. Then choose n_2 of x without the n_1 chosen elements beforehand and so on. You then get for example ({0,2},{},{1,3},{4}). Of course, every possible combination must be created so ({0,4},{},{1,3},{2}), too, and so on. Seems rather hard to implement but might be possible.

Research

I guess this goes in the direction I want however I don't see how I can utilize it for my specific scenario.

http://rosettacode.org/wiki/Combinations