A317792 Number of non-isomorphic multiset partitions, using normal multisets, of normal multisets of size n.

1, 1, 3, 6, 15, 31, 73, 154, 345, 742, 1627, 3499

Offset: 0

Author

Gus Wiseman, Aug 07 2018

Keywords

• nonn
• more

Comments

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if it has weakly decreasing multiplicities. Neither condition is necessarily preserved under isomorphism. For example, {{2},{1,1,1,2}} is isomorphic to {{1},{1,2,2,2}}, but only the latter has normal blocks, while only the former has strongly normal multiset union.

Examples

			Non-isomorphic representatives of the a(4) = 15 normal multiset partitions:
  {1111}, {1112}, {1122}, {1123}, {1234},
  {1}{111}, {1}{112}, {1}{122}, {1}{123}, {11}{11}, {11}{12}, {12}{12},
  {1}{1}{11}, {1}{1}{12},
  {1}{1}{1}{1}.
		

Crossrefs

Cf. A001055, A007716, A045778, A281116, A317533, A317757, A317791, A317794, A317795.

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]]]];
sysnorm[{}]:={};sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Union[sysnorm/@Select[Join@@mps/@allnorm[n],And@@(Union[#]==Range[Max@@#]&)/@#&]]],{n,6}]

Extensions

a(10)-a(11) from Robert Price, Sep 15 2018