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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.
13
1, 0, 1, 2, 5, 13, 33, 104, 293, 938, 2892
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
This is the case of A320804 where the underlying multiset is aperiodic.
Sequence in context: A052988 A001429 A148288 * A278134 A271940 A273721
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 08 2018
EXTENSIONS
Definition corrected by Gus Wiseman, Jan 19 2024
STATUS
approved