A000612 -id:A000612 - cp's OEIS frontend

A326898 Number of unlabeled topologies with up to n points.

1, 2, 5, 14, 47, 186, 904, 5439, 41418, 404501, 5122188, 84623842, 1828876351, 51701216248, 1908493827243, 91755916071736, 5729050033597431

Offset: 0

Gus Wiseman, Aug 02 2019

nonn

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 14 topologies:
  {}  {}     {}            {}
      {}{1}  {}{1}         {}{1}
             {}{12}        {}{12}
             {}{2}{12}     {}{123}
             {}{1}{2}{12}  {}{2}{12}
                           {}{3}{123}
                           {}{23}{123}
                           {}{1}{2}{12}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}

Partial sums of A001930.
The labeled version is A326878.
Cf. A000612, A000798, A003180, A108800, A193675, A306445, A326867, A326882.

A327335 Number of non-isomorphic set-systems with n vertices and at least one endpoint/leaf.

Original entry on oeis.org

0, 1, 4, 18, 216

Offset: 0

Gus Wiseman, Sep 02 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.

Also covering set-systems with minimum covered vertex-degree 1.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 18 set-systems:
  {{1}}  {{1}}        {{1}}
         {{1,2}}      {{1,2}}
         {{1},{2}}    {{1},{2}}
         {{1},{1,2}}  {{1,2,3}}
                      {{1},{1,2}}
                      {{1},{2,3}}
                      {{1},{1,2,3}}
                      {{1,2},{1,3}}
                      {{1},{2},{3}}
                      {{1,2},{1,2,3}}
                      {{1},{2},{1,3}}
                      {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{2},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1},{2},{3},{1,2}}
                      {{1},{2},{1,2},{1,3}}
                      {{1},{2},{1,2},{1,2,3}}

Unlabeled set-systems are A000612.
The labeled version is A327228.
The covering version is A327230 (first differences).
Cf. A002494, A245797, A261919, A283877, A327103, A327105, A327197, A327227, A327229, A327336.

A330294 Number of non-isomorphic fully chiral set-systems on n vertices.

Original entry on oeis.org

1, 2, 3, 10, 899

Offset: 0

Gus Wiseman, Dec 10 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 10 set-systems:
  0  0    0        0
     {1}  {1}      {1}
          {2}{12}  {2}{12}
                   {1}{3}{23}
                   {2}{13}{23}
                   {3}{23}{123}
                   {2}{3}{13}{23}
                   {1}{3}{23}{123}
                   {2}{13}{23}{123}
                   {2}{3}{13}{23}{123}

The labeled version is A330282.
Partial sums of A330295 (the covering case).
Unlabeled costrict (or T_0) set-systems are A326946.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Fully chiral partitions are A330228.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A016031, A055621, A083323, A283877, A319637, A330098, A330231, A330232, A330234.

A330295 Number of non-isomorphic fully chiral set-systems covering n vertices.

Original entry on oeis.org

1, 1, 1, 7, 889

Offset: 0

Gus Wiseman, Dec 10 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 7 set-systems:
  0  {1}  {1}{12}  {1}{2}{13}
                   {1}{12}{23}
                   {1}{12}{123}
                   {1}{2}{12}{13}
                   {1}{2}{13}{123}
                   {1}{12}{23}{123}
                   {1}{2}{12}{13}{123}

The labeled version is A330229.
First differences of A330294 (the non-covering case).
Unlabeled costrict (or T_0) set-systems are A326946.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Fully chiral partitions are A330228.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A016031, A055621, A083323, A283877, A319637, A330098, A330231, A330232, A330234, A330282.

A330344 Number of unlabeled graphs with n vertices whose covered portion has exactly two automorphisms.

Original entry on oeis.org

0, 1, 2, 4, 13, 50, 367

Offset: 1

Gus Wiseman, Dec 12 2019

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 graphs:
  {12}  {12}     {12}           {12}
        {12,13}  {12,13}        {12,13}
                 {12,13,24}     {12,13,24}
                 {12,13,14,23}  {12,13,14,23}
                                {12,13,14,25}
                                {12,13,24,35}
                                {12,13,14,23,25}
                                {12,13,14,23,45}
                                {12,13,15,24,34}
                                {12,13,14,15,23,24}
                                {12,13,14,23,24,35}
                                {12,13,14,23,25,45}
                                {12,13,14,15,23,24,35}

The labeled version is A330345.
The covering case is A330346 (not A241454).
Unlabeled graphs are A000088.
Unlabeled graphs with exactly one automorphism are A003400.
Unlabeled connected graphs with exactly one automorphism are A124059.
Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), and A330346 (unlabeled covering).
Cf. A000612, A004111, A055621, A241454, A283877, A330098, A330227, A330230, A330231, A330233, A330294, A330295.

Partial sums of A330346.

A368731 Number of non-isomorphic n-element sets of nonempty subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 10, 97, 2160, 126862, 21485262, 11105374322, 18109358131513, 95465831661532570, 1660400673336788987026, 96929369602251313489896310, 19268528295096123543660356281600, 13203875101002459910158494602665950757, 31517691852305548841992346407978317698725021

Offset: 0

Gus Wiseman, Jan 07 2024

nonn

			Non-isomorphic representatives of the a(3) = 10 set-systems:
  {{1},{2},{3}}
  {{1},{2},{1,2}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

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

The case of graphs is A001434, labeled A116508.
Labeled version is A136556, covering A054780, binomial transform of A367916.
The case of labeled covering graphs is A367863, binomial transform A367862.
These include the set-systems ranked by A367917.
The covering case is A368186, for graphs A006649, connected A057500.
Requiring all edges to be singletons or pairs gives A368598.
A003465 counts covers with any number of edges, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A007716, A283877, A367901, A367902, A367903, A368599.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Subsets[Subsets[Range[n],{1,n}],{n}]]],{n,0,4}]

a(n) = polcoef(G(n, n), n) \\ G defined in A368186. - Andrew Howroyd, Jan 11 2024

Terms a(6) and beyond from Andrew Howroyd, Jan 11 2024

A369202 Number of unlabeled simple graphs covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 0, 0, 2, 13, 95, 826, 11137, 261899, 11729360, 1006989636, 164072166301, 50336940172142, 29003653625802754, 31397431814146891910, 63969589218557753075156, 245871863137828405124380563, 1787331789281458167615190373076, 24636021675399858912682459601585276

Offset: 0

Gus Wiseman, Jan 23 2024

nonn

These are simple graphs covering n vertices such that some connected component has at least two cycles.

			Representatives of the a(4) = 2 and a(5) = 13 simple graphs:
  {12}{13}{14}{23}{24}      {12}{13}{14}{15}{23}{24}
  {12}{13}{14}{23}{24}{34}  {12}{13}{14}{15}{23}{45}
                            {12}{13}{14}{23}{24}{35}
                            {12}{13}{14}{23}{25}{45}
                            {12}{13}{14}{25}{35}{45}
                            {12}{13}{14}{15}{23}{24}{25}
                            {12}{13}{14}{15}{23}{24}{34}
                            {12}{13}{14}{15}{23}{24}{35}
                            {12}{13}{14}{23}{24}{35}{45}
                            {12}{13}{14}{15}{23}{24}{25}{34}
                            {12}{13}{14}{15}{23}{24}{35}{45}
                            {12}{13}{14}{15}{23}{24}{25}{34}{35}
                            {12}{13}{14}{15}{23}{24}{25}{34}{35}{45}

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

Without the choice condition we have A002494, labeled A006129.
The connected case is A140636.
This is the covering case of A140637, complement A134964.
The labeled version is A367868, complement A367869.
The complement is counted by A368834.
The version with loops is A369147, complement A369200.
A005703 counts unlabeled connected choosable simple graphs, labeled A129271.
A007716 counts unlabeled multiset partitions, connected A007718.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A283877 counts unlabeled set-systems, connected A300913.
Cf. A000088, A000612, A006649, A001434, A055621, A137916, A137917, A140638, A368596, A369141, A369146.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n] && Length[Select[Tuples[#],UnsameQ@@#&]]==0&]]],{n,0,5}]

First differences of A140637.

a(n) = A002494(n) - A368834(n).

A055152 Proper covers of an unlabeled n-set.

Original entry on oeis.org

0, 1, 14, 956, 9331320, 6406603065901952, 16879085743296493569230716352778240, 717956902513121252476003434439730211883694285987816199468264943161704448

Offset: 1

Vladeta Jovovic, Jun 14 2000

Alois P. Heinz, Table of n, a(n) for n = 1..12
Heller, Jürgen Identifiability in probabilistic knowledge structures. J. Math. Psychol. 77, 46-57 (2017).
Eric Weisstein's World of Mathematics, Proper covers

See A007537 for labeled case. Cf. A055621.

b:= proc(n, i, l) `if`(n=0, 2^(w-> add(mul(2^igcd(t, l[h]),
      h=1..nops(l)), t=1..w)/w)(ilcm(l[])), `if`(i<1, 0,
      add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i)))
    end:
a:= n->  (b(n$2, [])-2*b(n-1$2, []))/4:
seq(a(n), n=1..8);  # Alois P. Heinz, Aug 14 2019

b[n_] := Sum[1/Function[p, Product[Function[c, j^c*c!][Coefficient[p, x, j]], {j, 1, Exponent[p, x]}]][Total[x^l]]*2^(Function[w, Sum[Product[ 2^GCD[t, l[[i]]], {i, 1, Length[l]}], {t, 1, w}]/w][If[l == {}, 1, LCM @@ l]]), {l, IntegerPartitions[n]}];
a[n_] := (b[n] - 2 b[n - 1])/4;
a /@ Range[8] (* Jean-François Alcover, Feb 19 2020, after Alois P. Heinz in A000612 *)

a(n) = (A003180(n) - 2*A003180(n-1))/4.

Apparently a(n) = A002857(n) - A000612(n-1). - R. J. Mathar, Apr 22 2007

More terms from David Wasserman, Mar 21 2002

A326908 Number of non-isomorphic sets of subsets of {1..n} that are closed under union and intersection.

Original entry on oeis.org

2, 4, 9, 23, 70, 256, 1160, 6599, 48017, 452518, 5574706, 90198548, 1919074899, 53620291147, 1962114118390, 93718030190126, 5822768063787557

Offset: 0

Gus Wiseman, Aug 03 2019

			Non-isomorphic representatives of the a(0) = 2 through a(3) = 23 sets of subsets:
  {}    {}       {}              {}
  {{}}  {{}}     {{}}            {{}}
        {{1}}    {{1}}           {{1}}
        {{}{1}}  {{12}}          {{12}}
                 {{}{1}}         {{}{1}}
                 {{}{12}}        {{123}}
                 {{2}{12}}       {{}{12}}
                 {{}{2}{12}}     {{}{123}}
                 {{}{1}{2}{12}}  {{2}{12}}
                                 {{3}{123}}
                                 {{}{2}{12}}
                                 {{23}{123}}
                                 {{}{3}{123}}
                                 {{}{23}{123}}
                                 {{}{1}{2}{12}}
                                 {{3}{23}{123}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{3}{13}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

The labeled version is A306445.
Taking first differences and prepending 1 gives A326898.
Taking second differences and prepending two 1's gives A001930.
Cf. A000612, A000798, A003180, A108798, A108800, A193675, A326867, A326876, A326878, A326882, A326883.

Table[Length[Select[Subsets[Subsets[Range[n]]],SubsetQ[#,Union@@@Tuples[#,2]]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

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

cp's OEIS Frontend

A326898 Number of unlabeled topologies with up to n points.

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A327335 Number of non-isomorphic set-systems with n vertices and at least one endpoint/leaf.

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A330294 Number of non-isomorphic fully chiral set-systems on n vertices.

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A330295 Number of non-isomorphic fully chiral set-systems covering n vertices.

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A330344 Number of unlabeled graphs with n vertices whose covered portion has exactly two automorphisms.

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A368731 Number of non-isomorphic n-element sets of nonempty subsets of {1..n}.

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A369202 Number of unlabeled simple graphs covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

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A055152 Proper covers of an unlabeled n-set.

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A326908 Number of non-isomorphic sets of subsets of {1..n} that are closed under union and intersection.

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Mathematica

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

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