A371840 Number of integer partitions of n with non-biquanimous multiplicities.

0, 1, 2, 2, 4, 5, 8, 11, 16, 21, 31, 40, 55, 72, 97, 124, 165, 209, 271, 343, 441, 547, 700, 866, 1089, 1345, 1679, 2050, 2546, 3099, 3814, 4622, 5654, 6811, 8297, 9957, 12039, 14409, 17355, 20666, 24793, 29432, 35133, 41598, 49474, 58360, 69197, 81395, 96124

Offset: 0

Author

Gus Wiseman, Apr 18 2024

Keywords

• nonn

Comments

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.

Examples

			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).
The a(1) = 1 through a(8) = 16 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (321)     (421)      (422)
                            (11111)  (411)     (511)      (431)
                                     (3111)    (2221)     (521)
                                     (21111)   (4111)     (611)
                                     (111111)  (22111)    (2222)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
		

Crossrefs

The complement for parts is counted by A002219 aerated, ranks A357976.

These partitions have Heinz numbers A371782.

For parts we have A371795, ranks A371731, bisections A006827, A058695.

The complement is counted by A371839, ranks A371781.

A237258 = biquanimous strict partitions, ranks A357854, complement A371794.

A321451 counts non-quanimous partitions, ranks A321453.

A321452 counts quanimous partitions, ranks A321454.

A371783 counts k-quanimous partitions.

A371791 counts biquanimous sets, differences A232466.

A371792 counts non-biquanimous sets, differences A371793.

Cf. A035470, A064914, A305551, A321142, A365543, A365925, A367094.

Programs

Mathematica

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[IntegerPartitions[n], !biqQ[Length/@Split[#]]&]],{n,0,30}]