|
|
A371840
|
|
Number of integer partitions of n with non-biquanimous multiplicities.
|
|
4
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
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.
|
|
LINKS
|
|
|
EXAMPLE
|
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)
|
|
MATHEMATICA
|
biqQ[y_]:=MemberQ[Total/@Subsets[y], Total[y]/2];
Table[Length[Select[IntegerPartitions[n], !biqQ[Length/@Split[#]]&]], {n, 0, 30}]
|
|
CROSSREFS
|
The complement for parts is counted by A002219 aerated, ranks A357976.
These partitions have Heinz numbers A371782.
A371783 counts k-quanimous partitions.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|