Efficient Enumeration on m-combinations of the powerset of a set

Input:

s: A set of n elements, for example {1,2,3}

t: The power set of s, for example {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Output:

all sets: {u}, each u is an m-combination of t, and each element of u can not be a proper subset of another element, for example:

{{},{},{}}

{{1},{1},{1}}

{{1},{1},{2}}

{{1},{1},{3}}

{{1},{1},{23}}

{{1},{2},{2}}

{{1},{2},{3}}

{{1},{3},{3}}

{{1},{23},{23}}

{{2},{2},{2}}

{{2},{2},{3}}

{{2},{2},{13}}

{{2},{3},{3}}

{{2},{13},{13}}

{{2},{13},{23}}

{{3},{3},{3}}

{{3},{3},{12}}

{{3},{12},{12}}

{{12},{12},{12}}

{{12},{12},{13}}

{{12},{12},{23}}

{{12},{13},{13}}

{{12},{13},{23}}

{{12},{23},{23}}

{{13},{13},{13}}

{{13},{13},{23}}

{{23},{23},{23}}

{{123},{123},{123}}

The problem:

1 We do not know how many sets will be produced.

2 We hope for an efficient algorithm that produce all sets in a minimum steps.

Re: Efficient Enumeration on m-combinations of the powerset of a set

Have you tried an "inefficient" enumeration first?

It's always good to have an initial working solution even if it won't win you the Nobel price.