A371736 Number of non-quanimous strict integer partitions of n, meaning no set partition with more than one block has all equal block-sums.

1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 12, 11, 18, 15, 26, 23, 38, 30, 54, 43, 72, 57, 104, 77, 142, 102, 179, 138, 256, 170, 340, 232, 412, 292, 585, 365, 760, 471, 889, 602, 1260, 718, 1610, 935, 1819, 1148, 2590, 1371, 3264, 1733, 3581, 2137, 5120, 2485, 6372

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.

Examples

			The a(0) = 1 through a(9) = 8 strict partitions:
  ()  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)
                (21)  (31)  (32)  (42)  (43)   (53)   (54)
                            (41)  (51)  (52)   (62)   (63)
                                        (61)   (71)   (72)
                                        (421)  (521)  (81)
                                                      (432)
                                                      (531)
                                                      (621)
		

Crossrefs

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

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

The non-strict version is A321451, ranks A321453.

The complement is A371737, non-strict A321452, ranks A321454.

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

A108917 counts knapsack partitions, ranks A299702, strict A275972.

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

A371783 counts k-quanimous partitions.

A371789 counts non-quanimous sets, differences A371790.

A371792 counts non-biquanimous sets, complement A371791.

A371796 counts quanimous sets, differences A371797.

Cf. A000005, A018818, A035470, A038041, A232466, A279791, A365663, A365661, A365925, A371840.

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}]

Formula

a(prime(k)) = A064688(k) = A000009(A000040(k)).