A279375 Number of set partitions of strict integer partitions of n that have distinct block-sums.

1, 1, 1, 3, 3, 5, 9, 12, 16, 24, 39, 49, 70, 94, 127, 202, 247, 340, 450, 606, 772, 1169, 1407, 1920, 2454, 3267, 4089, 5469, 7293, 9222, 11884, 15291, 19417, 24890, 31469, 39662, 52619, 64764, 82502, 103576, 131169, 162726, 206015, 254233, 318464, 406262, 499210, 620593, 773673, 957073, 1181593

Offset: 0

Author

Gus Wiseman, Dec 11 2016

Keywords

• nonn

Comments

Also twice partitioned numbers where all partitions are strict. Also triangles of weight n in the multisystem of strict partitions. Strict partitions are an example of a multisystem that is neither transitive nor partitive nor contractible but is decomposable; see link for details.

Examples

			The a(6)=9 set partitions of strict integer partitions of 6 are: ((6)), ((51)), ((5)(1)), ((42)), ((4)(2)), ((321)), ((32)(1)), ((31)(2)), ((3)(2)(1)). The set partition ((3)(21)) is not counted because its blocks do not have distinct sums.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Gus Wiseman, Comcategories and Multiorders (pdf version)

Crossrefs

Cf. A063834, A089259, A258466, A270995, A271619, A275780, A279374.

Programs

Mathematica

nn=20;sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Total[Length[Select[sps[Sort[#]],UnsameQ@@Total/@#&]]&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,nn}]