OFFSET
0,4
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.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..100
Gus Wiseman, Comcategories and Multiorders (pdf version)
EXAMPLE
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.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 11 2016
STATUS
approved