A371789 Number of non-quanimous subsets of {1..n}, meaning there is only one set partition with all equal block-sums.

1, 2, 4, 7, 13, 24, 45, 85, 162, 306, 585, 1102, 2106, 3988, 7623, 14535, 27758, 52921, 101848, 195618, 378383, 733609, 1421868, 2755807, 5373060, 10482925, 20495335, 40119622, 78476107, 153463714, 300732073

Offset: 0

Author

Gus Wiseman, Apr 17 2024

Keywords

• nonn
• more

Comments

A finite multiset of numbers is defined to be quanimous iff it can be partitioned into two or more multisets with equal sums. Quanimous partitions are counted by A321452 and ranked by A321454.

Examples

			The set s = {3,4,6,8,9} has set partitions {{3,4,6,8,9}} and {{3,4,8},{6,9}} with equal block-sums, so s is not counted under a(9).
The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {2,3,4}
		

Crossrefs

The "bi-" complement for integer partitions is A002219, ranks A357976.

The "bi-" complement for strict partitions is A237258, ranks A357854.

The version for integer partitions is A321451, ranks A321453.

The complement for integer partitions is A321452, ranks A321454

The version for strict partitions is A371736, complement A371737.

First differences are A371790.

The "bi-" version is A371792, complement A371791.

The "bi-" version for strict partitions is A371794 (bisection A321142).

The "bi-" version for integer partitions is A371795, ranks A371731.

The complement is counted by A371796, differences A371797.

A108917 counts knapsack partitions, ranks A299702, strict A275972.

A366754 counts non-knapsack partitions, ranks A299729, strict A316402.

A371783 counts k-quanimous partitions.

Cf. A000005, A018818, A035470, A038041, A232466, A365663, A365661.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Subsets[Range[n]], Length[Select[sps[#],SameQ@@Total/@#&]]==1&]],{n,0,8}]

Extensions

a(11)-a(30) from Bert Dobbelaere, Mar 30 2025