A320813 Number of non-isomorphic multiset partitions of an aperiodic multiset of weight n such that there are no singletons and all parts are themselves aperiodic multisets.

1, 0, 1, 2, 5, 13, 33, 104, 293, 938, 2892

Offset: 0

Author

Gus Wiseman, Nov 08 2018

Keywords

• nonn
• more

Comments

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which (1) the row sums are all > 1, (2) the positive entries in each row are relatively prime, and (3) the column-sums are relatively prime.

A multiset is aperiodic if its multiplicities are relatively prime.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 multiset partitions:
  {{1,2}}  {{1,2,2}}  {{1,2,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,3,3}}    {{1,2,2,2,2}}
                      {{1,2,3,4}}    {{1,2,2,3,3}}
                      {{1,2},{3,4}}  {{1,2,3,3,3}}
                      {{1,3},{2,3}}  {{1,2,3,4,4}}
                                     {{1,2,3,4,5}}
                                     {{1,2},{1,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,3},{1,2,3}}
		

Crossrefs

This is the case of A320804 where the underlying multiset is aperiodic.

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A302545, A303386, A303546, A303707, A303708, A320797-A320813, A321283, A321390.

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]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
aperQ[m_]:=Length[m]==0||GCD@@Length/@Split[Sort[m]]==1;
Table[Length[Union[brute /@ Select[mpm[n],And[Min@@Length/@#>1,aperQ[Join@@#]&&And@@aperQ /@ #]&]]],{n,0,7}] (* Gus Wiseman, Jan 19 2024 *)

Extensions

Definition corrected by Gus Wiseman, Jan 19 2024