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

0,11

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.

Links

Table of n, a(n) for n=0..60.

Example

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)

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}]

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.

Sequence in context: A066705 A277894 A027636 * A173425 A289445 A237558

Adjacent sequences: A371734 A371735 A371736 * A371738 A371739 A371740

Keyword

nonn

Author

Gus Wiseman, Apr 14 2024

Status

approved