OFFSET
0,4
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.
EXAMPLE
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}}
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 *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 08 2018
EXTENSIONS
Definition corrected by Gus Wiseman, Jan 19 2024
STATUS
approved