A326173 Number of maximal subsets of {1..n} whose sum is less than or equal to the sum of their complement.

1, 1, 1, 2, 4, 5, 8, 16, 24, 44, 77, 133, 240, 429, 772, 1414, 2588, 4742, 8761, 16273, 30255, 56392, 105581, 198352, 373228, 703409, 1329633, 2519927, 4781637, 9084813, 17298255, 33001380, 63023204, 120480659, 230702421, 442423139, 849161669, 1631219288, 3137595779, 6042247855, 11644198080, 22455871375, 43351354727

Offset: 0

Author

Gus Wiseman, Jun 11 2019

Keywords

• nonn

Comments

Also the number of minimal subsets of {1..n} whose sum is greater than or equal to the sum of their complement. For example, the a(0) = 1 through a(7) = 16 subsets are:

{} {1} {2} {3} {1,4} {3,5} {5,6} {1,6,7}

{1,2} {2,3} {4,5} {1,4,6} {2,5,7}

{2,4} {1,2,5} {2,3,6} {2,6,7}

{3,4} {1,3,4} {2,4,5} {3,4,7}

{2,3,4} {2,4,6} {3,5,6}

{3,4,5} {3,5,7}

{3,4,6} {3,6,7}

{1,2,3,5} {4,5,6}

{4,5,7}

{4,6,7}

{5,6,7}

{1,2,4,7}

{1,2,5,6}

{1,3,4,6}

{2,3,4,5}

{2,3,4,6}

Examples

			The a(0) = 1 through a(7) = 16 subsets:
  {}  {}  {1}  {3}    {1,2}  {1,5}    {4,6}      {1,5,7}
               {1,2}  {1,3}  {2,5}    {1,2,5}    {1,6,7}
                      {1,4}  {3,4}    {1,2,6}    {2,5,7}
                      {2,3}  {1,2,3}  {1,3,5}    {3,4,7}
                             {1,2,4}  {1,3,6}    {3,5,6}
                                      {1,4,5}    {1,2,3,4}
                                      {2,3,5}    {1,2,3,5}
                                      {1,2,3,4}  {1,2,3,6}
                                                 {1,2,3,7}
                                                 {1,2,4,5}
                                                 {1,2,4,6}
                                                 {1,2,4,7}
                                                 {1,2,5,6}
                                                 {1,3,4,5}
                                                 {1,3,4,6}
                                                 {2,3,4,5}
		

Crossrefs

The non-maximal case is A059529.

Cf. A053632, A057567, A057568, A063865, A326174, A326175.

Programs

Mathematica

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],Plus@@Complement[Range[n],#]>=Plus@@#&]]],{n,0,10}]

Extensions

a(16)-a(42) from Bert Dobbelaere, Jun 22 2019