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%I #5 Apr 08 2024 09:14:15
%S 1,1,1,2,4,8,18,38,82,175,373,787,1651,3439,7126,14667
%N Number of biquanimous subsets of {1..n}. Sets with a subset having the same sum as the complement.
%C A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.
%e For S = {1,3,4,6} we have {{1,6},{3,4}}, so S is counted under a(6).
%e The a(0) = 1 through a(6) = 18 subsets:
%e {} {} {} {} {} {} {}
%e {1,2,3} {1,2,3} {1,2,3} {1,2,3}
%e {1,3,4} {1,3,4} {1,3,4}
%e {1,2,3,4} {1,4,5} {1,4,5}
%e {2,3,5} {1,5,6}
%e {1,2,3,4} {2,3,5}
%e {1,2,4,5} {2,4,6}
%e {2,3,4,5} {1,2,3,4}
%e {1,2,3,6}
%e {1,2,4,5}
%e {1,2,5,6}
%e {1,3,4,6}
%e {2,3,4,5}
%e {2,3,5,6}
%e {3,4,5,6}
%e {1,2,3,4,6}
%e {1,2,4,5,6}
%e {2,3,4,5,6}
%t biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
%t Table[Length[Select[Subsets[Range[n]],biqQ]],{n,0,15}]
%Y First differences are A232466.
%Y The complement is counted by A371792, differences A371793.
%Y This is the "bi-" case of A371796, differences A371797.
%Y A002219 aerated counts biquanimous partitions, ranks A357976.
%Y A006827 and A371795 count non-biquanimous partitions, ranks A371731.
%Y A108917 counts knapsack partitions, ranks A299702, strict A275972.
%Y A237258 aerated counts biquanimous strict partitions, ranks A357854.
%Y A321142 and A371794 count non-biquanimous strict partitions.
%Y A321451 counts non-quanimous partitions, ranks A321453.
%Y A321452 counts quanimous partitions, ranks A321454.
%Y A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
%Y A371737 counts quanimous strict partitions, complement A371736.
%Y A371781 lists numbers with biquanimous prime signature, complement A371782.
%Y A371783 counts k-quanimous partitions.
%Y Cf. A035470, A064914, A357879, A365661, A366320, A365381, A365925, A367094, A371788, A371789.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Apr 07 2024