A302545 -id:A302545 - cp's OEIS frontend

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

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

A330053 Number of non-isomorphic set-systems of weight n with at least one singleton.

Original entry on oeis.org

0, 1, 1, 3, 6, 14, 32, 79, 193, 499, 1321, 3626, 10275, 30126, 91062, 284093, 912866, 3018825, 10261530, 35814255, 128197595, 470146011, 1764737593, 6773539331, 26561971320, 106330997834, 434195908353, 1807306022645, 7663255717310, 33079998762373

Offset: 0

Gus Wiseman, Nov 30 2019

nonn

A set-system is a finite set of finite nonempty sets of positive integers. An singleton is an edge of size 1. The weight of a set-system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 14 multiset partitions:
  {1}  {1}{2}  {1}{12}    {1}{123}      {1}{1234}
               {1}{23}    {1}{234}      {1}{2345}
               {1}{2}{3}  {1}{2}{12}    {1}{12}{13}
                          {1}{2}{13}    {1}{12}{23}
                          {1}{2}{34}    {1}{12}{34}
                          {1}{2}{3}{4}  {1}{2}{123}
                                        {1}{2}{134}
                                        {1}{2}{345}
                                        {1}{23}{45}
                                        {2}{13}{14}
                                        {1}{2}{3}{12}
                                        {1}{2}{3}{14}
                                        {1}{2}{3}{45}
                                        {1}{2}{3}{4}{5}

Jean-François Alcover, Table of n, a(n) for n = 0..50 [using data from A283877 and A306005]

The complement is counted by A306005.
The multiset partition version is A330058.
Non-isomorphic set-systems with at least one endpoint are A330052.
Non-isomorphic set-systems counted by vertices are A000612.
Non-isomorphic set-systems counted by weight are A283877.
Cf. A007716, A055621, A302545, A317533, A317794, A319559, A320665, A330055, A330056, A330057.

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A283877 = A@283877;
A306005 = A@306005;
a[n_] := A283877[[n + 1]] - A306005[[n + 1]];
a /@ Range[0, 50] (* Jean-François Alcover, Feb 09 2020 *)

a(n) = A283877(n) - A306005(n). - Jean-François Alcover, Feb 09 2020

A330059 Number of set-systems with n vertices and no endpoints.

Original entry on oeis.org

1, 1, 2, 63, 29471, 2144945976, 9223371624669871587, 170141183460469227599616678821978424151, 57896044618658097711785492504343953752410420469299789800819363538011879603532

Offset: 0

Gus Wiseman, Dec 01 2019

nonn

A set-system is a finite set of finite nonempty set of positive integers. An endpoint is a vertex appearing only once (degree 1).

			The a(2) = 2 set-systems are {} and {{1},{2},{1,2}}. The a(3) = 63 set-systems are:
  0                 {2}{3}{12}{13}       {1}{3}{12}{13}{23}
  {1}{2}{12}        {2}{12}{13}{23}      {2}{3}{12}{13}{23}
  {1}{3}{13}        {2}{3}{12}{123}      {1}{2}{12}{23}{123}
  {2}{3}{23}        {2}{3}{13}{123}      {1}{2}{13}{23}{123}
  {12}{13}{23}      {3}{12}{13}{23}      {1}{3}{12}{13}{123}
  {1}{23}{123}      {1}{13}{23}{123}     {1}{3}{12}{23}{123}
  {2}{13}{123}      {2}{12}{13}{123}     {1}{3}{13}{23}{123}
  {3}{12}{123}      {2}{12}{23}{123}     {2}{3}{12}{13}{123}
  {12}{13}{123}     {2}{13}{23}{123}     {2}{3}{12}{23}{123}
  {12}{23}{123}     {3}{12}{13}{123}     {2}{3}{13}{23}{123}
  {13}{23}{123}     {3}{12}{23}{123}     {1}{12}{13}{23}{123}
  {1}{2}{13}{23}    {3}{13}{23}{123}     {2}{12}{13}{23}{123}
  {1}{2}{3}{123}    {12}{13}{23}{123}    {3}{12}{13}{23}{123}
  {1}{3}{12}{23}    {1}{2}{3}{12}{13}    {1}{2}{3}{12}{13}{23}
  {1}{12}{13}{23}   {1}{2}{3}{12}{23}    {1}{2}{3}{12}{13}{123}
  {1}{2}{13}{123}   {1}{2}{3}{13}{23}    {1}{2}{3}{12}{23}{123}
  {1}{2}{23}{123}   {1}{2}{12}{13}{23}   {1}{2}{3}{13}{23}{123}
  {1}{3}{12}{123}   {1}{2}{3}{12}{123}   {1}{2}{12}{13}{23}{123}
  {1}{3}{23}{123}   {1}{2}{3}{13}{123}   {1}{3}{12}{13}{23}{123}
  {1}{12}{13}{123}  {1}{2}{3}{23}{123}   {2}{3}{12}{13}{23}{123}
  {1}{12}{23}{123}  {1}{2}{12}{13}{123}  {1}{2}{3}{12}{13}{23}{123}

Andrew Howroyd, Table of n, a(n) for n = 0..11

The case with no singletons is A330056.
The unlabeled version is A330054 (by weight) or A330124 (by vertices).
Set-systems with no singletons are A016031.
Non-isomorphic set-systems with no singletons are A306005 (by weight).
Cf. A000612, A007716, A055621, A302545, A317533, A317794, A319559, A320665, A321405, A330052, A330057, A330058.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Min@@Length/@Split[Sort[Join@@#]]>1&]],{n,0,4}]

a(n) = {sum(k=0, n, (-1)^k*binomial(n,k)*2^(2^(n-k)-1)*sum(j=0, k, stirling(k,j,2)*2^(j*(n-k)) ))} \\ Andrew Howroyd, Jan 16 2023

a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^k * binomial(n,k) * 2^(2^(n-k)-1) * Stirling2(k,j) * 2^(j*(n-k)). - Andrew Howroyd, Jan 16 2023

Terms a(5) and beyond from Andrew Howroyd, Jan 16 2023

A320295 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

0, 1, 2, 5, 8, 19, 34, 80, 165, 394, 892, 2192, 5232, 13057, 32271, 81568, 205748, 525735, 1344828, 3467415, 8960849, 23280323, 60639680, 158559047, 415631368, 1092734050, 2879420753, 7605713020, 20130266302, 53386744298, 141836904569, 377479973474, 1006189769886

Offset: 1

Gus Wiseman, Oct 09 2018

nonn

Also phylogenetic trees with no singleton leaves on integer partitions of n.

			The a(2) = 1 through a(6) = 19 trees:
  (11)  (21)   (22)        (32)         (33)
        (111)  (31)        (41)         (42)
               (211)       (221)        (51)
               (1111)      (311)        (222)
               ((11)(11))  (2111)       (321)
                           (11111)      (411)
                           ((11)(12))   (2211)
                           ((11)(111))  (3111)
                                        (21111)
                                        (111111)
                                        ((11)(13))
                                        ((11)(22))
                                        ((12)(12))
                                        ((11)(112))
                                        ((12)(111))
                                        ((11)(1111))
                                        ((111)(111))
                                        ((11)(11)(11))
                                        ((11)((11)(11)))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000311, A000669, A005804, A141268, A302545, A304966, A319312, A320289, A320294.

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]]]];
pgtm[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[pgtm/@p]],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[Select[pgtm[m],FreeQ[#,{_}]&]],{m,IntegerPartitions[n]}],{n,14}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=1, n, 1 - x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, n) - 1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(12) and beyond from Andrew Howroyd, Oct 25 2018

A321406 Number of non-isomorphic self-dual set systems of weight n with no singletons.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 1, 2, 4

Offset: 0

Gus Wiseman, Nov 15 2018

Also the number of 0-1 symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the rows are all different and none sums to 1.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
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(6) = 1 through a(10) = 4 set systems:
   6: {{1,2},{1,3},{2,3}}
   7: {{1,3},{2,3},{1,2,3}}
   8: {{1,2},{1,3},{2,4},{3,4}}
   9: {{1,2},{1,3},{1,4},{2,3,4}}
   9: {{1,2},{1,4},{3,4},{2,3,4}}
  10: {{1,2},{2,4},{1,3,4},{2,3,4}}
  10: {{1,3},{2,4},{1,3,4},{2,3,4}}
  10: {{1,4},{2,4},{3,4},{1,2,3,4}}
  10: {{1,2},{1,3},{2,4},{3,5},{4,5}}

Cf. A000219, A007716, A045778, A049311, A135588, A138178, A283877, A302545, A316980, A316983.

Cf. A320796, A320797, A321401, A321402, A321403, A321405.

A321411 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons, with aperiodic parts whose sizes are relatively prime.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 4, 6, 16, 25

Offset: 0

Gus Wiseman, Nov 16 2018

A multiset is aperiodic if its multiplicities are relatively prime.
Also the number of nonnegative integer symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with relatively prime row sums (or column sums) and no row or column having a common divisor > 1 or summing to 1.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
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(5) = 1 through a(9) = 16 multiset partitions:
  {{12}{122}}  {{112}{1222}}    {{112}{12222}}    {{1112}{11222}}
               {{12}{12222}}    {{122}{11222}}    {{1112}{12222}}
               {{12}{13}{233}}  {{12}{123}{233}}  {{12}{1222222}}
               {{13}{23}{123}}  {{13}{112}{233}}  {{12}{123}{2333}}
                                {{13}{122}{233}}  {{12}{13}{23333}}
                                {{23}{123}{123}}  {{12}{223}{1233}}
                                                  {{13}{112}{2333}}
                                                  {{13}{223}{1233}}
                                                  {{13}{23}{12333}}
                                                  {{23}{122}{1233}}
                                                  {{23}{123}{1233}}
                                                  {{12}{12}{34}{234}}
                                                  {{12}{12}{34}{344}}
                                                  {{12}{13}{14}{234}}
                                                  {{12}{13}{24}{344}}
                                                  {{12}{14}{34}{234}}

Cf. A000219, A007716, A120733, A138178, A302545, A316983, A319616.

Cf. A320796, A320797, A320803, A320806, A320809, A320813, A321283, A321408-A321413.

A320799 Number of non-isomorphic (not necessarily strict) antichains of multisets of weight n with no singletons or leaves (vertices that appear only once).

Original entry on oeis.org

1, 0, 1, 1, 5, 4, 22, 27, 107, 212, 689

Offset: 0

Gus Wiseman, Nov 02 2018

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(7) = 27 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{1122}}    {{11222}}    {{111222}}      {{1112222}}
                   {{11}{11}}  {{11}{122}}  {{112222}}      {{1122222}}
                   {{11}{22}}  {{11}{222}}  {{112233}}      {{1122333}}
                   {{12}{12}}               {{111}{111}}    {{111}{1222}}
                                            {{11}{1222}}    {{11}{12222}}
                                            {{111}{222}}    {{111}{2222}}
                                            {{112}{122}}    {{11}{12233}}
                                            {{11}{2222}}    {{111}{2233}}
                                            {{112}{222}}    {{112}{1222}}
                                            {{11}{2233}}    {{11}{22222}}
                                            {{112}{233}}    {{112}{2222}}
                                            {{122}{122}}    {{11}{22333}}
                                            {{123}{123}}    {{112}{2333}}
                                            {{11}{11}{11}}  {{113}{2233}}
                                            {{11}{12}{22}}  {{122}{1233}}
                                            {{11}{22}{22}}  {{222}{1122}}
                                            {{11}{22}{33}}  {{11}{11}{122}}
                                            {{11}{23}{23}}  {{11}{11}{222}}
                                            {{12}{12}{12}}  {{11}{12}{222}}
                                            {{12}{12}{22}}  {{11}{12}{233}}
                                            {{12}{13}{23}}  {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{12}{12}{222}}
                                                            {{12}{12}{233}}
                                                            {{12}{12}{333}}
                                                            {{12}{13}{233}}

Cf. A006126, A096827, A285572, A293993, A293994, A302545, A318099, A319560, A319719, A319721.

Cf. A320797, A320798, A320804, A320811, A320812, A320813.

A321404 Number of non-isomorphic self-dual set multipartitions (multisets of sets) of weight n with no singletons.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 1, 3, 4, 6

Offset: 0

Gus Wiseman, Nov 15 2018

Also the number of 0-1 symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which no row sums to 1.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
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(4) = 1 through a(10) = 6 set multipartitions:
   4: {{1,2},{1,2}}
   6: {{1,2},{1,3},{2,3}}
   7: {{1,3},{2,3},{1,2,3}}
   8: {{2,3},{1,2,3},{1,2,3}}
   8: {{1,2},{1,2},{3,4},{3,4}}
   8: {{1,2},{1,3},{2,4},{3,4}}
   9: {{1,2,3},{1,2,3},{1,2,3}}
   9: {{1,2},{1,2},{3,4},{2,3,4}}
   9: {{1,2},{1,3},{1,4},{2,3,4}}
   9: {{1,2},{1,4},{3,4},{2,3,4}}
  10: {{1,2},{1,2},{1,3,4},{2,3,4}}
  10: {{1,2},{2,4},{1,3,4},{2,3,4}}
  10: {{1,3},{2,4},{1,3,4},{2,3,4}}
  10: {{1,4},{2,4},{3,4},{1,2,3,4}}
  10: {{1,2},{1,2},{3,4},{3,5},{4,5}}
  10: {{1,2},{1,3},{2,4},{3,5},{4,5}}

Cf. A007716, A049311, A135588, A138178, A283877, A302545, A316983.

Cf. A320797, A320798, A320811, A320812, A321403, A321404, A321405, A321406.

A330057 Number of set-systems covering n vertices with no singletons or endpoints.

Original entry on oeis.org

1, 0, 0, 5, 1703, 66954642, 144115175199102143, 1329227995784915808340204290157341181, 226156424291633194186662080095093568664788471116325389572604136316742486364

Offset: 0

Gus Wiseman, Nov 30 2019

nonn

A set-system is a finite set of finite nonempty set of positive integers. A singleton is an edge of size 1. An endpoint is a vertex appearing only once (degree 1).

			The a(3) = 5 set-systems:
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}
  {{1,2},{2,3},{1,2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..11
Wikipedia, Degree (graph theory)

The version for non-isomorphic set-systems is A330055 (by weight).
The non-covering version is A330056.
Set-systems with no singletons are A016031.
Set-systems with no endpoints are A330059.
Non-isomorphic set-systems with no singletons are A306005 (by weight).
Non-isomorphic set-systems with no endpoints are A330054 (by weight).
Non-isomorphic set-systems counted by vertices are A000612.
Non-isomorphic set-systems counted by weight are A283877.
Cf. A007716, A055621, A302545, A317533, A317794, A319559, A320665, A321405, A330052, A330058.

Table[Length[Select[Subsets[Subsets[Range[n],{2,n}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]>1&]],{n,0,4}]

\\ here b(n) is A330056(n).
AS2(n, k) = {sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) )}
b(n) = {sum(k=0, n, (-1)^k*binomial(n,k)*2^(2^(n-k)-(n-k)-1) * sum(j=0, k\2, sum(i=0, k-2*j, binomial(k,i) * AS2(k-i, j) * (2^(n-k)-1)^i * 2^(j*(n-k)) )))}
a(n) = {sum(k=0, n, (-1)^k*binomial(n,k)*b(n-k))} \\ Andrew Howroyd, Jan 16 2023

Binomial transform is A330056.

Terms a(5) and beyond from Andrew Howroyd, Jan 16 2023

A321402 Number of non-isomorphic strict self-dual multiset partitions of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 8, 14, 27, 53, 105

Offset: 0

Gus Wiseman, Nov 09 2018

Also the number of nonnegative integer symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the rows are all different and none sums to 1.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
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(7) = 14 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{11}{22}}  {{11}{122}}  {{111}{222}}    {{111}{1222}}
                               {{11}{222}}  {{112}{122}}    {{111}{2222}}
                               {{12}{122}}  {{11}{2222}}    {{112}{1222}}
                                            {{12}{1222}}    {{11}{22222}}
                                            {{22}{1122}}    {{12}{12222}}
                                            {{11}{22}{33}}  {{122}{1122}}
                                            {{12}{13}{23}}  {{22}{11222}}
                                                            {{11}{12}{233}}
                                                            {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{11}{23}{233}}
                                                            {{12}{13}{233}}
                                                            {{13}{23}{123}}

Cf. A000219, A007716, A059201, A302545, A316980, A316983, A319560, A319616.

Cf. A320796, A320797, A320798, A320804, A320811, A320812, A321401, A321404, A321405, A321406, A321407.

cp's OEIS Frontend

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

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A330053 Number of non-isomorphic set-systems of weight n with at least one singleton.

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A330059 Number of set-systems with n vertices and no endpoints.

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A320295 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n.

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A321406 Number of non-isomorphic self-dual set systems of weight n with no singletons.

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A321411 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons, with aperiodic parts whose sizes are relatively prime.

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A320799 Number of non-isomorphic (not necessarily strict) antichains of multisets of weight n with no singletons or leaves (vertices that appear only once).

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A321404 Number of non-isomorphic self-dual set multipartitions (multisets of sets) of weight n with no singletons.

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A330057 Number of set-systems covering n vertices with no singletons or endpoints.

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