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

Offset

1,4

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.

Links

Table of n, a(n) for n=1..10.

Example

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}

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}]

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.

Sequence in context: A336482 A134389 A286945 * A350326 A111297 A077864

Adjacent sequences: A371794 A371795 A371796 * A371798 A371799 A371800

Keyword

nonn,more

Author

Gus Wiseman, Apr 17 2024

Status

approved