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# Revision history [back]

First, if your want to filter (non-necessary maximal) subsets with a that condition implies testing all subsets of your given set A, you can try something like:

sage: A = Set([0,1,2,3,4,5])
sage: B = Set(a for a in A.subsets() if len(a) != 0 and max(a) - min(a) <= 3)


Note that i had to exclude the empty set to be able to pick the minimal and the maximal element of the iterated subsets.

Then, you could notice that your property is stable by subsets, so you have more than a set of subsets, you have a simplicial complex:

sage: SimplicialComplex(B)
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and facets {(1, 2, 3, 4), (2, 3, 4, 5), (0, 1, 2, 3)}


From which you can get maximal faces:

sage: C = SimplicialComplex(B).maximal_faces() ; C
{(1, 2, 3, 4), (2, 3, 4, 5), (0, 1, 2, 3)}


In your second example, you get:

sage: A = Set([0,3,9,10,17,30,33])
sage: B = Set(a for a in A.subsets() if len(a) != 0 and max(a) - min(a) <= 3)
sage: C = SimplicialComplex(B).maximal_faces() ; C
{(9, 10), (17,), (30, 33), (0, 3)}


First, if your want to filter (non-necessary maximal) subsets with a that condition that implies testing all subsets of your given set A, you can try something like:

sage: A = Set([0,1,2,3,4,5])
sage: B = Set(a for a in A.subsets() if len(a) != 0 and max(a) - min(a) <= 3)


Note that i had to exclude the empty set to be able to pick the minimal and the maximal element of the iterated subsets of A. Note that i hd to use a Sage Set and not a python set to be able to get its subsets.

Then, you could notice that your property is stable by subsets, so you have more than a set of subsets, you have a simplicial complex:

sage: SimplicialComplex(B)
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and facets {(1, 2, 3, 4), (2, 3, 4, 5), (0, 1, 2, 3)}


From which you can get maximal faces:

sage: C = SimplicialComplex(B).maximal_faces() ; C
{(1, 2, 3, 4), (2, 3, 4, 5), (0, 1, 2, 3)}


In your second example, you get:

sage: A = Set([0,3,9,10,17,30,33])
sage: B = Set(a for a in A.subsets() if len(a) != 0 and max(a) - min(a) <= 3)
sage: C = SimplicialComplex(B).maximal_faces() ; C
{(9, 10), (17,), (30, 33), (0, 3)}