A384350 Number of subsets of {1..n} containing at least one element that is a sum of distinct non-elements.

0, 0, 0, 1, 4, 13, 33, 81, 183, 402, 856, 1801, 3721, 7646, 15567, 31575

Offset: 0

Author

Gus Wiseman, Jun 05 2025

Keywords

• nonn
• more

Comments

Conjecture: Also the number of subsets of {1..n} such that it is possible in more than one way to choose a disjoint family of strict integer partitions, one of each element.

Examples

			For the set s = {1,5} we have 5 = 2+3, so s is counted under a(5).
The a(0) = 0 through a(5) = 13 subsets:
  .  .  .  {3}  {3}    {3}
                {4}    {4}
                {2,4}  {5}
                {3,4}  {1,5}
                       {2,4}
                       {2,5}
                       {3,4}
                       {3,5}
                       {4,5}
                       {1,4,5}
                       {2,3,5}
                       {2,4,5}
                       {3,4,5}
		

Crossrefs

The complement is counted by A326080, allowing repeats A326083.

For strict partitions of n instead of subsets of {1..n} we have A384318, ranks A384322.

First differences are A384391.

A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.

A179009 counts maximally refined strict partitions, ranks A383707.

A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.

A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.

A383706 counts ways to choose disjoint strict partitions of prime indices, non-disjoint A357982, non-strict A299200.

Cf. A279375, A279790, A317141, A317142, A383708, A383710, A384317, A384319, A384320, A384321.

Programs

Mathematica

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@nonsets[#]]!={}&]],{n,0,10}]