A371794 - OEIS

%I #6 Apr 08 2024 09:13:58

%S 0,1,1,2,2,3,3,5,5,8,7,12,11,18,15,27,23,38,30,54,43,76,57,104,79,142,

%T 102,192,138,256,174,340,232,448,292,585,375,760,471,982,602,1260,741,

%U 1610,935,2048,1148,2590,1425,3264,1733,4097,2137,5120,2571,6378

%N Number of non-biquanimous strict 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(11) = 12 strict partitions:

%e   (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)

%e             (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)   (65)

%e                         (41)  (51)  (52)   (62)   (63)   (73)   (74)

%e                                     (61)   (71)   (72)   (82)   (83)

%e                                     (421)  (521)  (81)   (91)   (92)

%e                                                   (432)  (631)  (A1)

%e                                                   (531)  (721)  (542)

%e                                                   (621)         (632)

%e                                                                 (641)

%e                                                                 (731)

%e                                                                 (821)

%e                                                                 (5321)

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

%t Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&!biqQ[#]&]],{n,0,30}]

%Y The complement is counted by A237258 aerated, ranks A357854.

%Y Even bisection is A321142, odd A078408.

%Y This is the "bi-" version of A371736, complement A371737.

%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 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 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. A064914, A279787, A305551, A318434, A365543, A365663, A365661, A366320, A365925, A367094, A371788.

%K nonn

%O 0,4

%A _Gus Wiseman_, Apr 07 2024