A371839 Number of integer partitions of n with biquanimous multiplicities.

1, 0, 0, 1, 1, 2, 3, 4, 6, 9, 11, 16, 22, 29, 38, 52, 66, 88, 114, 147, 186, 245, 302, 389, 486, 613, 757, 960, 1172, 1466, 1790, 2220, 2695, 3332, 4013, 4926, 5938, 7228, 8660, 10519, 12545, 15151, 18041, 21663, 25701, 30774, 36361, 43359, 51149, 60720, 71374

Offset

0,6

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

Table of n, a(n) for n=0..50.

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 counted under a(10).
The a(0) = 1 through a(10) = 11 partitions:
  ()  .  .  (21)  (31)  (32)  (42)    (43)    (53)    (54)      (64)
                        (41)  (51)    (52)    (62)    (63)      (73)
                              (2211)  (61)    (71)    (72)      (82)
                                      (3211)  (3221)  (81)      (91)
                                              (3311)  (3321)    (3322)
                                              (4211)  (4221)    (4321)
                                                      (4311)    (4411)
                                                      (5211)    (5221)
                                                      (222111)  (5311)
                                                                (6211)
                                                                (322111)

Mathematica

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

Crossrefs

For parts instead of multiplicities we have A002219 aerated, ranks A357976.

These partitions have Heinz numbers A371781.

The complement for parts instead of multiplicities is counted by A371795, ranks A371731, bisections A006827, A058695.

The complement is counted by A371840, ranks A371782.

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.

Sequence in context: A007210 A198394 A035947 * A338914 A048249 A370843

Adjacent sequences: A371836 A371837 A371838 * A371840 A371841 A371842

Keyword

nonn

Author

Gus Wiseman, Apr 18 2024

Status

approved