A316972 Number of connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.

1, 2, 5, 28, 277, 3985, 76117, 1833187, 53756682, 1871041538, 75809298105, 3521419837339, 185235838688677, 10923147890901151, 715989783027216302, 51793686238309903860, 4109310551278549543317, 355667047514571431358297, 33422937748872646130124797

Offset: 0

Author

Gus Wiseman, Jul 17 2018

Keywords

• nonn

Comments

Note that all connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} are strict except for (123...n)(123...n).

Examples

			The a(2) = 5 connected multiset partitions of {1, 1, 2, 2} are (1122), (1)(122), (2)(112), (12)(12), (1)(2)(12). The multiset partitions (11)(22), (1)(1)(22), (2)(2)(11), (1)(1)(2)(2) are not connected.
		

Crossrefs

Cf. A001055, A007716, A007717, A007719, A020555, A045778, A061742, A094574, A162247, A316974.

Programs

Mathematica

nn=10;
ser=Exp[-3/2+Exp[x]/2]*Sum[Exp[Binomial[n+1,2]*x]/n!,{n,0,3*nn}];
Round/@(CoefficientList[Series[1+Log[ser],{x,0,nn}],x]*Array[Factorial,nn+1,0]) (* based on Jean-François Alcover after Vladeta Jovovic *)
(*second program *)
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]]],Union@@s[[c[[1]]]]]]]]];
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]]]];
Length/@Table[Select[mps[Ceiling[Range[1/2,n,1/2]]],Length[csm[#]]==1&],{n,4}]

Formula

Logarithmic transform of A020555.