A383308 Number of integer partitions of n that can be partitioned into sets with a common sum.

1, 1, 2, 3, 4, 4, 8, 6, 10, 13, 15, 13, 31

Offset: 0

Author

Gus Wiseman, Apr 25 2025

Keywords

• nonn
• more

Comments

Any strict partition can be partitioned into a single set, so we have a lower bound a(n) >= A000009(n).

Examples

			The multiset (3,2,2,1,1) has partition {{3},{1,2},{1,2}}, so is counted under a(9).
The a(1) = 1 through a(9) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)         (9)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)        (54)
             (111)  (31)    (41)     (42)      (52)       (53)        (63)
                    (1111)  (11111)  (51)      (61)       (62)        (72)
                                     (222)     (421)      (71)        (81)
                                     (321)     (1111111)  (431)       (333)
                                     (2211)               (521)       (432)
                                     (111111)             (2222)      (531)
                                                          (3311)      (621)
                                                          (11111111)  (3321)
                                                                      (32211)
                                                                      (222111)
                                                                      (111111111)
		

Crossrefs

Twice-partitions of this type (into sets with a common sum) are counted by A279788.

Multiset partitions of this type are ranked by A326534 /\ A302478.

For distinct instead of equal sums we have A381992, see also A382077.

The complement is counted by A381994, ranks A381719.

Partitions of prime indices of this type are counted by A382080.

Normal multiset partitions of this type are counted by A382429, see A326518.

For constant instead of strict blocks we have A383093, ranks A383014.

A000041 counts integer partitions, strict A000009.

A001055 counts factorizations, strict A045778.

Set multipartitions: A089259, A116540, A270995, A296119, A318360.

Cf. A050320, A293511, A321455, A358914, A381633, A381717, A381993.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#],And@@UnsameQ@@@#&&SameQ@@Total/@#&]]>0&]],{n,0,10}]