A381994 Number of integer partitions of n that cannot be partitioned into sets with equal sums.

0, 0, 0, 0, 1, 3, 3, 9, 12, 17, 27, 43, 46, 82, 103, 133, 181, 258, 295

Offset: 0

Author

Gus Wiseman, Mar 17 2025

Keywords

• nonn

Examples

			For y = (3,3,1,1) we have {{1,3},{1,3}}, so y is not counted under a(8).
For y = (3,2,2,1), although we have {{1,3},{2,2}}, the block {2,2} is not a set, so y is counted under a(8).
The a(4) = 1 through a(8) = 12 partitions:
  (2,1,1)  (2,2,1)    (4,1,1)      (3,2,2)        (3,3,2)
           (3,1,1)    (3,1,1,1)    (3,3,1)        (4,2,2)
           (2,1,1,1)  (2,1,1,1,1)  (5,1,1)        (6,1,1)
                                   (2,2,2,1)      (3,2,2,1)
                                   (3,2,1,1)      (4,2,1,1)
                                   (4,1,1,1)      (5,1,1,1)
                                   (2,2,1,1,1)    (2,2,2,1,1)
                                   (3,1,1,1,1)    (3,2,1,1,1)
                                   (2,1,1,1,1,1)  (4,1,1,1,1)
                                                  (2,2,1,1,1,1)
                                                  (3,1,1,1,1,1)
                                                  (2,1,1,1,1,1,1)
		

Crossrefs

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.

Twice-partitions of this type are counted by A279788.

Interchanging "constant" with "strict" gives A381717, see A381635, A381636, A381991.

Normal multiset partitions of this type are counted by A381718, see A279785.

These partitions are ranked by A381719, zeros of A382080.

For distinct instead of equal block-sums we have A381990, ranked by A381806.

For constant instead of strict blocks we have A381993.

A000041 counts integer partitions, strict A000009.

A050320 counts factorizations into squarefree numbers, see A381078, A381454.

A050326 counts factorizations into distinct squarefree numbers.

A265947 counts refinement-ordered pairs of integer partitions.

A381633 counts set systems with distinct sums, see A381634, A293243.

Cf. A002846, A047966, A279786, A279789, A293511, A299202, A317142, A358914, A381992.

Programs

Mathematica

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}]