Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #6 Apr 08 2024 09:14:04
%S 1,2,3,6,12,22,44,84,163,314,610,1184,2308,4505,8843
%N Number of non-biquanimous subsets of {1..n} containing n.
%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 The a(1) = 1 through a(5) = 12 subsets:
%e {1} {2} {3} {4} {5}
%e {1,2} {1,3} {1,4} {1,5}
%e {2,3} {2,4} {2,5}
%e {3,4} {3,5}
%e {1,2,4} {4,5}
%e {2,3,4} {1,2,5}
%e {1,3,5}
%e {2,4,5}
%e {3,4,5}
%e {1,2,3,5}
%e {1,3,4,5}
%e {1,2,3,4,5}
%t biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
%t Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&!biqQ[#]&]],{n,15}]
%Y The complement is counted by A232466, differences of A371791.
%Y This is the "bi-" version of A371790, differences of A371789.
%Y First differences of A371792.
%Y The complement is the "bi-" version of A371797, differences of A371796.
%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, A365543, A365661, A365663, A366320, A365381, A367094, A371788.
%K nonn,more
%O 1,2
%A _Gus Wiseman_, Apr 07 2024