A371797 Number of quanimous subsets of {1..n} containing n, meaning there is more than one set partition with equal block-sums.

0, 0, 1, 2, 5, 11, 24, 51, 112, 233, 507, 1044, 2214, 4557, 9472, 19545, 40373, 82145, 168374, 341523, 693350, 1408893, 2860365, 5771355, 11667351, 23542022, 47484577, 95861243, 193447849, 389602553

Offset: 1

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 counted under a(9).
The a(1) = 0 through a(6) = 11 subsets:
  .  .  {1,2,3}  {1,3,4}    {1,4,5}      {1,5,6}
                 {1,2,3,4}  {2,3,5}      {2,4,6}
                            {1,2,4,5}    {1,2,3,6}
                            {2,3,4,5}    {1,2,5,6}
                            {1,2,3,4,5}  {1,3,4,6}
                                         {2,3,5,6}
                                         {3,4,5,6}
                                         {1,2,3,4,6}
                                         {1,2,4,5,6}
                                         {2,3,4,5,6}
                                         {1,2,3,4,5,6}
		

Crossrefs

The "bi-" version is A232466, complement A371793.

The complement is counted by A371790.

First differences of A371796, complement A371789.

A371736 counts non-quanimous strict partitions.

A371737 counts quanimous strict partitions.

A371783 counts k-quanimous partitions.

A371791 counts biquanimous subsets, complement A371792.

Cf. A000005, A002219, A035470, A038041, A275972, A279791, A321452, A371795.

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]], MemberQ[#,n]&&Length[Select[sps[#],SameQ@@Total/@#&]]>1&]],{n,10}]

Extensions

a(11)-a(30) from Martin Fuller, Apr 01 2025