A371737 Number of quanimous strict integer partitions of n, meaning there is more than one set partition with all equal block-sums.

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 4, 0, 7, 1, 9, 0, 16, 0, 21, 4, 32, 0, 45, 0, 63, 13, 84, 0, 126, 0, 158, 36, 220, 0, 303, 0, 393, 93, 511, 0, 708, 0, 881, 229, 1156, 0, 1539, 0, 1925, 516, 2445, 0, 3233, 6, 3952, 1134, 5019, 0, 6497

Offset: 0

Author

Gus Wiseman, Apr 14 2024

Keywords

• nonn

Comments

A finite multiset of numbers is defined to be quanimous iff it can be partitioned into two or more multisets with equal sums. Quanimous partitions are counted by A321452 and ranked by A321454.

Conjecture: (1) Positions of 0's are A327782. (2) Positions of terms > 0 are A368459.

Examples

			The a(0) = 0 through a(14) = 7 strict partitions:
  .  .  .  .  .  .  (321)  .  (431)  .  (532)   .  (642)   .  (743)
                                        (541)      (651)      (752)
                                        (4321)     (5421)     (761)
                                                   (6321)     (5432)
                                                              (6431)
                                                              (6521)
                                                              (7421)
		

Crossrefs

The non-strict "bi-" version is A002219, ranks A357976.

The "bi-" version is A237258, ranks A357854, complement A321142 or A371794.

The non-strict version is A321452, ranks A321454.

The complement is A371736, non-strict A321451, ranks A321453.

The non-strict "bi-" complement is A371795, ranks A371731.

A371783 counts k-quanimous partitions.

A371791 counts biquanimous sets, complement A371792.

A371796 counts quanimous sets, complement A371789.

Cf. A000005, A018818, A035470, A038041, A064688, A232466, A237194, A305551, A365663, A365661, A365925, A371733, A371839.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[Select[sps[#], SameQ@@Total/@#&]]>1&]],{n,0,30}]