This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A371840 #5 Apr 18 2024 09:32:39
%S A371840 0,1,2,2,4,5,8,11,16,21,31,40,55,72,97,124,165,209,271,343,441,547,
%T A371840 700,866,1089,1345,1679,2050,2546,3099,3814,4622,5654,6811,8297,9957,
%U A371840 12039,14409,17355,20666,24793,29432,35133,41598,49474,58360,69197,81395,96124
%N A371840 Number of integer partitions of n with non-biquanimous multiplicities.
%C A371840 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 A371840 The partition y = (6,2,1,1) has multiplicities (1,1,2), which are biquanimous because we have the partition ((1,1),(2)), so y is not counted under a(10).
%e A371840 The a(1) = 1 through a(8) = 16 partitions:
%e A371840 (1) (2) (3) (4) (5) (6) (7) (8)
%e A371840 (11) (111) (22) (221) (33) (322) (44)
%e A371840 (211) (311) (222) (331) (332)
%e A371840 (1111) (2111) (321) (421) (422)
%e A371840 (11111) (411) (511) (431)
%e A371840 (3111) (2221) (521)
%e A371840 (21111) (4111) (611)
%e A371840 (111111) (22111) (2222)
%e A371840 (31111) (5111)
%e A371840 (211111) (22211)
%e A371840 (1111111) (32111)
%e A371840 (41111)
%e A371840 (221111)
%e A371840 (311111)
%e A371840 (2111111)
%e A371840 (11111111)
%t A371840 biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
%t A371840 Table[Length[Select[IntegerPartitions[n], !biqQ[Length/@Split[#]]&]],{n,0,30}]
%Y A371840 The complement for parts is counted by A002219 aerated, ranks A357976.
%Y A371840 These partitions have Heinz numbers A371782.
%Y A371840 For parts we have A371795, ranks A371731, bisections A006827, A058695.
%Y A371840 The complement is counted by A371839, ranks A371781.
%Y A371840 A237258 = biquanimous strict partitions, ranks A357854, complement A371794.
%Y A371840 A321451 counts non-quanimous partitions, ranks A321453.
%Y A371840 A321452 counts quanimous partitions, ranks A321454.
%Y A371840 A371783 counts k-quanimous partitions.
%Y A371840 A371791 counts biquanimous sets, differences A232466.
%Y A371840 A371792 counts non-biquanimous sets, differences A371793.
%Y A371840 Cf. A035470, A064914, A305551, A321142, A365543, A365925, A367094.
%K A371840 nonn
%O A371840 0,3
%A A371840 _Gus Wiseman_, Apr 18 2024