A371794 Number of non-biquanimous strict integer partitions of n.

0, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 12, 11, 18, 15, 27, 23, 38, 30, 54, 43, 76, 57, 104, 79, 142, 102, 192, 138, 256, 174, 340, 232, 448, 292, 585, 375, 760, 471, 982, 602, 1260, 741, 1610, 935, 2048, 1148, 2590, 1425, 3264, 1733, 4097, 2137, 5120, 2571, 6378

Offset: 0

Author

Gus Wiseman, Apr 07 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 a(1) = 1 through a(11) = 12 strict partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)   (65)
                        (41)  (51)  (52)   (62)   (63)   (73)   (74)
                                    (61)   (71)   (72)   (82)   (83)
                                    (421)  (521)  (81)   (91)   (92)
                                                  (432)  (631)  (A1)
                                                  (531)  (721)  (542)
                                                  (621)         (632)
                                                                (641)
                                                                (731)
                                                                (821)
                                                                (5321)
		

Crossrefs

The complement is counted by A237258 aerated, ranks A357854.

Even bisection is A321142, odd A078408.

This is the "bi-" version of A371736, complement A371737.

A002219 aerated counts biquanimous partitions, ranks A357976.

A006827 and A371795 count non-biquanimous partitions, ranks A371731.

A108917 counts knapsack partitions, ranks A299702, strict A275972.

A321451 counts non-quanimous partitions, ranks A321453.

A321452 counts quanimous partitions, ranks A321454.

A366754 counts non-knapsack partitions, ranks A299729, strict A316402.

A371781 lists numbers with biquanimous prime signature, complement A371782.

A371783 counts k-quanimous partitions.

A371789 counts non-quanimous sets, differences A371790.

A371791 counts biquanimous sets, differences A232466.

A371792 counts non-biquanimous sets, differences A371793.

A371796 counts quanimous sets, differences A371797.

Cf. A064914, A279787, A305551, A318434, A365543, A365663, A365661, A366320, A365925, A367094, A371788.

Programs

Mathematica

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