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

0,2

Comments

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

Links

Table of n, a(n) for n=0..18.

Formula

Logarithmic transform of A020555.

Example

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.

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}]

Crossrefs

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

Sequence in context: A342288 A324264 A138293 * A068069 A292499 A306893

Adjacent sequences: A316969 A316970 A316971 * A316973 A316974 A316975

Keyword

nonn

Author

Gus Wiseman, Jul 17 2018

Status

approved