A317078 Number of connected multiset partitions of strongly normal multisets of size n.

1, 1, 3, 6, 18, 46, 172, 563, 2347

Offset: 0

Author

Gus Wiseman, Jul 20 2018

Keywords

• nonn
• more

Comments

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

Examples

			The a(3) = 6 connected multiset partitions are (111), (1)(11), (1)(1)(1), (112), (1)(12), (123).
		

Crossrefs

Cf. A007716, A007718, A048143, A293994, A303837, A303838, A304716, A305078.

Cf. A317074, A317076, A317077, A317080.

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]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Length/@Table[Join@@Table[Select[mps[m],Length[csm[#]]==1&],{m,strnorm[n]}],{n,8}]