A368727 Number of non-isomorphic connected multiset partitions of weight n into singletons or strict pairs.
1, 1, 2, 2, 5, 6, 15, 21, 49, 82, 184, 341, 766, 1530, 3428, 7249, 16394, 36009, 82492, 186485, 433096, 1001495, 2358182, 5554644, 13255532, 31718030, 76656602, 185982207, 454889643, 1117496012, 2764222322, 6868902152, 17172601190
Offset: 0
Keywords
Examples
Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 multiset partitions:
{1} {12} {2}{12} {12}{12} {2}{12}{12} {12}{12}{12}
{1}{1} {1}{1}{1} {13}{23} {2}{13}{23} {12}{13}{23}
{1}{2}{12} {3}{13}{23} {13}{23}{23}
{2}{2}{12} {1}{2}{2}{12} {13}{24}{34}
{1}{1}{1}{1} {2}{2}{2}{12} {14}{24}{34}
{1}{1}{1}{1}{1} {1}{2}{12}{12}
{1}{2}{13}{23}
{2}{2}{12}{12}
{2}{2}{13}{23}
{2}{3}{13}{23}
{3}{3}{13}{23}
{1}{1}{2}{2}{12}
{1}{2}{2}{2}{12}
{2}{2}{2}{2}{12}
{1}{1}{1}{1}{1}{1}
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..50
Crossrefs
This is the connected case of A339888.
Programs
-
Mathematica
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}]; mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}]; csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List /@ c[[1]]],Union@@s[[c[[1]]]]]]]]]; brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]]; Table[Length[Union[brute /@ Select[mpm[n],And@@UnsameQ@@@#&&Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]
Formula
Inverse Euler transform of A339888.