cp's OEIS Frontend

A371839 Number of integer partitions of n with biquanimous multiplicities.

%I A371839 #7 Apr 18 2024 09:32:52
%S A371839 1,0,0,1,1,2,3,4,6,9,11,16,22,29,38,52,66,88,114,147,186,245,302,389,
%T A371839 486,613,757,960,1172,1466,1790,2220,2695,3332,4013,4926,5938,7228,
%U A371839 8660,10519,12545,15151,18041,21663,25701,30774,36361,43359,51149,60720,71374
%N A371839 Number of integer partitions of n with biquanimous multiplicities.
%C A371839 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 A371839 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 counted under a(10).
%e A371839 The a(0) = 1 through a(10) = 11 partitions:
%e A371839   ()  .  .  (21)  (31)  (32)  (42)    (43)    (53)    (54)      (64)
%e A371839                         (41)  (51)    (52)    (62)    (63)      (73)
%e A371839                               (2211)  (61)    (71)    (72)      (82)
%e A371839                                       (3211)  (3221)  (81)      (91)
%e A371839                                               (3311)  (3321)    (3322)
%e A371839                                               (4211)  (4221)    (4321)
%e A371839                                                       (4311)    (4411)
%e A371839                                                       (5211)    (5221)
%e A371839                                                       (222111)  (5311)
%e A371839                                                                 (6211)
%e A371839                                                                 (322111)
%t A371839 biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
%t A371839 Table[Length[Select[IntegerPartitions[n], biqQ[Length/@Split[#]]&]],{n,0,30}]
%Y A371839 For parts instead of multiplicities we have A002219 aerated, ranks A357976.
%Y A371839 These partitions have Heinz numbers A371781.
%Y A371839 The complement for parts instead of multiplicities is counted by A371795, ranks A371731, bisections A006827, A058695.
%Y A371839 The complement is counted by A371840, ranks A371782.
%Y A371839 A237258 = biquanimous strict partitions, ranks A357854, complement A371794.
%Y A371839 A321451 counts non-quanimous partitions, ranks A321453.
%Y A371839 A321452 counts quanimous partitions, ranks A321454.
%Y A371839 A371783 counts k-quanimous partitions.
%Y A371839 A371791 counts biquanimous sets, differences A232466.
%Y A371839 A371792 counts non-biquanimous sets, differences A371793.
%Y A371839 Cf. A035470, A064914, A305551, A321142, A365543, A365925, A367094.
%K A371839 nonn
%O A371839 0,6
%A A371839 _Gus Wiseman_, Apr 18 2024