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Number of minimal complete subsets of {1..n} with maximum n.
3

%I #12 Jun 03 2024 13:16:44

%S 1,1,1,1,2,2,2,4,8,8,8,10,14,25,40,49,62

%N Number of minimal complete subsets of {1..n} with maximum n.

%C A set of positive integers summing to m is complete if every nonnegative integer up to m is the sum of some subset. For example, (1,2,3,6,13) is a complete set because we have:

%C 0 = (empty sum)

%C 1 = 1

%C 2 = 2

%C 3 = 3

%C 4 = 1 + 3

%C 5 = 2 + 3

%C 6 = 6

%C 7 = 6 + 1

%C 8 = 6 + 2

%C 9 = 6 + 3

%C 10 = 1 + 3 + 6

%C 11 = 2 + 3 + 6

%C 12 = 1 + 2 + 3 + 6

%C and the remaining numbers 13-25 are obtained by adding 13 to each of these.

%H Andrzej Kukla and Piotr Miska, <a href="https://arxiv.org/abs/2405.18225">On practical sets and A-practical numbers</a>, arXiv:2405.18225 [math.NT], 2024.

%e The a(3) = 1 through a(9) = 8 subsets:

%e {1,2,3} {1,2,4} {1,2,3,5} {1,2,3,6} {1,2,3,7} {1,2,4,8} {1,2,3,4,9}

%e {1,2,4,5} {1,2,4,6} {1,2,4,7} {1,2,3,5,8} {1,2,3,5,9}

%e {1,2,3,6,8} {1,2,3,6,9}

%e {1,2,3,7,8} {1,2,3,7,9}

%e {1,2,4,5,9}

%e {1,2,4,6,9}

%e {1,2,4,7,9}

%e {1,2,4,8,9}

%t fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]&/@Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];

%t Table[Length[fasmin[Select[Subsets[Range[n]],Max@@#==n&&Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]]],{n,10}]

%Y Cf. A002033, A103295, A108917, A126796, A188431, A276024.

%Y Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326020, A326021, A326036.

%K nonn,more

%O 1,5

%A _Gus Wiseman_, Jun 04 2019