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Number of non-biquanimous subsets of {1..n} containing n.
14

%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