%I #6 Apr 08 2024 09:13:51

%S 0,1,1,3,2,7,5,15,8,30,17,56,24,101,46,176,64,297,107,490,147,792,242,

%T 1255,302,1958,488,3010,629,4565,922

%N Number of non-biquanimous integer partitions of 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(8) = 8 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (21) (31) (32) (42) (43) (53)

%e (111) (41) (51) (52) (62)

%e (221) (222) (61) (71)

%e (311) (411) (322) (332)

%e (2111) (331) (521)

%e (11111) (421) (611)

%e (511) (5111)

%e (2221)

%e (3211)

%e (4111)

%e (22111)

%e (31111)

%e (211111)

%e (1111111)

%t biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];

%t Table[Length[Select[IntegerPartitions[n],Not@*biqQ]],{n,0,15}]

%Y The complement is counted by A002219 aerated, ranks A357976.

%Y Even bisection is A006827, odd A058695.

%Y The strict complement is A237258, ranks A357854.

%Y This is the "bi-" version of A321451, ranks A321453.

%Y The complement is the "bi-" version of A321452, ranks A321454.

%Y These partitions have ranks A371731.

%Y The strict case is A371794, bisections A321142, A078408.

%Y A108917 counts knapsack partitions, ranks A299702, strict A275972.

%Y A366754 counts non-knapsack partitions, ranks A299729, strict A316402.

%Y A371736 counts non-quanimous strict partitons, complement A371737.

%Y A371781 lists numbers with biquanimous prime signature, complement A371782.

%Y A371783 counts k-quanimous partitions.

%Y A371789 counts non-quanimous sets, differences A371790.

%Y A371791 counts biquanimous sets, differences A232466.

%Y A371792 counts non-biquanimous sets, differences A371793.

%Y A371796 counts quanimous sets, differences A371797.

%Y Cf. A035470, A064914, A305551, A336137, A365543, A365661, A365663, A366320, A365925, A367094, A371788.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Apr 07 2024