A064914 Number of ordered biquanimous partitions of 2n.

1, 1, 5, 23, 105, 449, 1902, 7828, 31976, 129200, 520425, 2088217, 8371186, 33514797, 134140430, 536699674, 2147154667, 8589198795, 34358341823, 137435830265, 549749857574, 2199010044813, 8796067657649, 35184315676573, 140737380485376, 562949713881526

Offset: 0

Author

Christian G. Bower, Oct 12 2001

Keywords

• nonn

Comments

A biquanimous partition is one that can be bisected into two equal sized parts: e.g. 3+2+1 is a biquanimous partition of 6 as it contains 3 and 2+1, but 5+1 is not.

Examples

			From _Gus Wiseman_, Apr 19 2024: (Start)
The a(0) = 1 through a(3) = 23 biquanimous compositions:
  ()  (11)  (22)    (33)
            (112)   (123)
            (121)   (132)
            (211)   (213)
            (1111)  (231)
                    (312)
                    (321)
                    (1113)
                    (1122)
                    (1131)
                    (1212)
                    (1221)
                    (1311)
                    (2112)
                    (2121)
                    (2211)
                    (3111)
                    (11112)
                    (11121)
                    (11211)
                    (12111)
                    (21111)
                    (111111)
(End)
		

Crossrefs

The unordered version (integer partitions) is A002219, ranks A357976.

The unordered complement is A371795, even case A006827, ranks A371731.

The complement is counted by A371956.

These compositions have ranks A372120, complement A372119.

A237258 (aerated) counts biquanimous strict partitions, ranks A357854.

A321142 and A371794 count non-biquanimous strict partitions.

A371791 counts biquanimous sets, differences A232466.

A371792 counts non-biquanimous sets, differences A371793.

Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[2n], MemberQ[Total/@Subsets[#],n]&]],{n,0,5}] (* Gus Wiseman, Apr 19 2024 *)

Extensions

More terms from Alois P. Heinz, Jun 12 2017