A321390 -id:A321390 - cp's OEIS frontend

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

Gus Wiseman, Nov 08 2018

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.

			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}}

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.

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 *)

Definition corrected by Gus Wiseman, Jan 19 2024

A320811 Number of non-isomorphic multiset partitions with no singletons of aperiodic multisets of size n.

Original entry on oeis.org

1, 0, 1, 2, 7, 21, 57, 200, 575, 1898, 5893

Offset: 0

Gus Wiseman, Nov 08 2018

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 and (2) 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.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 21 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},{2,2}}  {{1,2,3,3,3}}
                      {{1,2},{3,3}}  {{1,2,3,4,4}}
                      {{1,2},{3,4}}  {{1,2,3,4,5}}
                      {{1,3},{2,3}}  {{1,1},{1,2,2}}
                                     {{1,1},{2,2,2}}
                                     {{1,1},{2,3,3}}
                                     {{1,1},{2,3,4}}
                                     {{1,2},{1,2,2}}
                                     {{1,2},{2,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,2},{1,2,2}}
                                     {{2,3},{1,2,3}}
                                     {{3,3},{1,2,3}}

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

A320812 Number of non-isomorphic aperiodic multiset partitions of weight n with no singletons.

Original entry on oeis.org

1, 0, 2, 3, 10, 23, 79, 204, 670, 1974, 6521, 21003, 71944, 248055, 888565, 3240552, 12152093, 46527471, 182337383, 729405164, 2979114723, 12407307929, 52670334237, 227725915268, 1002285201807, 4487915293675, 20434064047098, 94559526594316, 444527729321513

Offset: 0

Gus Wiseman, Nov 08 2018

nonn

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.

			Non-isomorphic representatives of the a(2) = 2 through a(5) = 23 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}
                      {{1,1},{2,2}}  {{1,2,3,4,4}}
                      {{1,2},{2,2}}  {{1,2,3,4,5}}
                      {{1,2},{3,3}}  {{1,1},{1,1,1}}
                      {{1,2},{3,4}}  {{1,1},{1,2,2}}
                      {{1,3},{2,3}}  {{1,1},{2,2,2}}
                                     {{1,1},{2,3,3}}
                                     {{1,1},{2,3,4}}
                                     {{1,2},{1,2,2}}
                                     {{1,2},{2,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,2},{1,2,2}}
                                     {{2,3},{1,2,3}}
                                     {{3,3},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

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

a(n) = Sum_{d|n} mu(d)*A302545(n/d) for n > 0. - Andrew Howroyd, Jan 16 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

A323584 Second Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic.

Original entry on oeis.org

1, 1, 1, 4, 8, 22, 34, 84, 137, 271, 450, 857, 1373, 2483, 3993, 6823, 10990, 18332, 28966, 47328, 74286, 118614, 184755, 290781, 448010, 695986, 1063773, 1632100, 2474970, 3759610, 5654233, 8512307, 12710995, 18973247, 28139285, 41690830, 61423271, 90379782

Offset: 0

Gus Wiseman, Jan 19 2019

nonn

A multiset is aperiodic if its multiplicities are relatively prime.

Also the number of plane partitions of n whose multiset of rows is aperiodic and whose parts are relatively prime.

			The a(4) = 8 plane partitions with aperiodic multisets of rows and columns:
  4   31   211
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1
The a(4) = 8 plane partitions with aperiodic multiset of rows and relatively prime parts:
  31   211   1111
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000219, A000837, A003293, A100953, A300275, A303546, A320802, A321390, A323585, A323587.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[GCD@@Length/@Split[#]==1,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],GCD@@#==1&]}],{n,10}]

The Moebius transform T of a sequence q is T(q)(n) = Sum_{d|n} mu(n/d) * q(d) where mu = A008683. The first Moebius transform of A000219 is A300275 and the third is A323585.

A323585 Third Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic and whose parts are relatively prime.

Original entry on oeis.org

1, 1, 0, 3, 7, 21, 30, 83, 129, 267, 428, 856, 1332, 2482, 3909, 6798, 10853, 18331, 28665, 47327, 73829, 118527, 183898, 290780, 446508, 695964, 1061290, 1631829, 2470970, 3759609, 5646952, 8512306, 12700005, 18972387, 28120953, 41690725, 61392966, 90379781

Offset: 0

Gus Wiseman, Jan 19 2019

nonn

A multiset is aperiodic if its multiplicities are relatively prime.

			The a(4) = 7 plane partitions with aperiodic multisets of rows and columns and relatively prime parts:
  31   211
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1
The same for a(5) = 21:
  41   32   311   221   2111
.
  4   3   31   21   22   21   211   111   1111
  1   2   1    2    1    11   1     11    1
.
  3   2   21   11   111
  1   2   1    11   1
  1   1   1    1    1
.
  2   11
  1   1
  1   1
  1   1

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000219, A000837, A003293, A100953, A300275, A303546, A320802, A321390, A323584, A323587.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[GCD@@Length/@Split[#]==1,GCD@@Length/@Split[Transpose[PadRight[#]]]==1,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],GCD@@#==1&]}],{n,10}]

The Moebius transform T of a sequence q is T(q)(n) = Sum_{d|n} mu(n/d) * q(d) where mu = A008683. The first Moebius transform of A000219 is A300275 and the second is A323584.

cp's OEIS Frontend

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.

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A320811 Number of non-isomorphic multiset partitions with no singletons of aperiodic multisets of size n.

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A320812 Number of non-isomorphic aperiodic multiset partitions of weight n with no singletons.

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A323584 Second Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic.

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A323585 Third Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic and whose parts are relatively prime.

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