cp's OEIS Frontend

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

%I A371797 #9 Apr 01 2025 08:55:55
%S A371797 0,0,1,2,5,11,24,51,112,233,507,1044,2214,4557,9472,19545,40373,82145,
%T A371797 168374,341523,693350,1408893,2860365,5771355,11667351,23542022,
%U A371797 47484577,95861243,193447849,389602553
%N A371797 Number of quanimous subsets of {1..n} containing n, meaning there is more than one set partition with equal block-sums.
%C A371797 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.
%e A371797 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).
%e A371797 The a(1) = 0 through a(6) = 11 subsets:
%e A371797   .  .  {1,2,3}  {1,3,4}    {1,4,5}      {1,5,6}
%e A371797                  {1,2,3,4}  {2,3,5}      {2,4,6}
%e A371797                             {1,2,4,5}    {1,2,3,6}
%e A371797                             {2,3,4,5}    {1,2,5,6}
%e A371797                             {1,2,3,4,5}  {1,3,4,6}
%e A371797                                          {2,3,5,6}
%e A371797                                          {3,4,5,6}
%e A371797                                          {1,2,3,4,6}
%e A371797                                          {1,2,4,5,6}
%e A371797                                          {2,3,4,5,6}
%e A371797                                          {1,2,3,4,5,6}
%t A371797 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A371797 Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&Length[Select[sps[#],SameQ@@Total/@#&]]>1&]],{n,10}]
%Y A371797 The "bi-" version is A232466, complement A371793.
%Y A371797 The complement is counted by A371790.
%Y A371797 First differences of A371796, complement A371789.
%Y A371797 A371736 counts non-quanimous strict partitions.
%Y A371797 A371737 counts quanimous strict partitions.
%Y A371797 A371783 counts k-quanimous partitions.
%Y A371797 A371791 counts biquanimous subsets, complement A371792.
%Y A371797 Cf. A000005, A002219, A035470, A038041, A275972, A279791, A321452, A371795.
%K A371797 nonn,more
%O A371797 1,4
%A A371797 _Gus Wiseman_, Apr 17 2024
%E A371797 a(11)-a(30) from _Martin Fuller_, Apr 01 2025