A000612 -id:A000612 - cp's OEIS frontend

A368836 Triangle read by rows where T(n,k) is the number of unlabeled loop-graphs on up to n vertices with k loops and n-k non-loops.

1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 2, 6, 6, 2, 1, 6, 17, 18, 8, 2, 1, 21, 52, 58, 30, 9, 2, 1, 65, 173, 191, 107, 37, 9, 2, 1, 221, 585, 666, 393, 148, 39, 9, 2, 1, 771, 2064, 2383, 1493, 589, 168, 40, 9, 2, 1, 2769, 7520, 8847, 5765, 2418, 718, 176, 40, 9, 2, 1

Offset: 0

Gus Wiseman, Jan 11 2024

Are the row sums the same as column k = 1 (shifted left)?

Yes. When k = 1 there is one loop. Remove the vertex with the loop and add loops to its neighbors. This process is reversible so there is a bijection. - Andrew Howroyd, Jan 13 2024

			Triangle begins:
   1
   0  1
   0  1  1
   1  2  2  1
   2  6  6  2  1
   6 17 18  8  2  1
  21 52 58 30  9  2  1
Representatives of the loop-graphs counted by row n = 4:
  {12}{13}{14}{23} {1}{12}{13}{14} {1}{2}{12}{13} {1}{2}{3}{12} {1}{2}{3}{4}
  {12}{13}{24}{34} {1}{12}{13}{23} {1}{2}{12}{34} {1}{2}{3}{14}
                   {1}{12}{13}{24} {1}{2}{13}{14}
                   {1}{12}{23}{24} {1}{2}{13}{23}
                   {1}{12}{23}{34} {1}{2}{13}{24}
                   {1}{23}{24}{34} {1}{2}{13}{34}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Column k = 0 is A001434.
Row sums are A368598.
The labeled version is A368928.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts loop-graphs, unlabeled A000666.
A058891 counts set-systems, unlabeled A000612.
Cf. A000272, A007717, A062740, A322661, A333331, A368596, A368730, A368835.

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],{1,2}],{n}],Count[#,{_}]==k&]]], {n,0,4},{k,0,n}]

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
row(n) = {my(s=0, A=1+O(x*x^n)); forpart(p=n, s+=permcount(p) * polcoef(edges(p, i->A + x^i)*prod(i=1, #p, A + (x*y)^p[i]), n)); Vecrev(s/n!)} \\ Andrew Howroyd, Jan 13 2024

a(28) onwards from Andrew Howroyd, Jan 13 2024

A369200 Number of unlabeled loop-graphs covering n vertices such that it is possible to choose a different vertex from each edge (choosable).

Original entry on oeis.org

1, 1, 3, 7, 18, 43, 112, 282, 740, 1940, 5182, 13916, 37826, 103391, 284815, 788636, 2195414, 6137025, 17223354, 48495640, 136961527, 387819558, 1100757411, 3130895452, 8922294498, 25470279123, 72823983735, 208515456498, 597824919725, 1716072103910, 4931540188084

Offset: 0

Gus Wiseman, Jan 23 2024

nonn

These are covering loop-graphs with at most one cycle (unicyclic) in each connected component.

			Representatives of the a(1) = 1 through a(4) = 18 loop-graphs (loops shown as singletons):
  {{1}}  {{1,2}}      {{1},{2,3}}          {{1,2},{3,4}}
         {{1},{2}}    {{1,2},{1,3}}        {{1},{2},{3,4}}
         {{1},{1,2}}  {{1},{2},{3}}        {{1},{1,2},{3,4}}
                      {{1},{2},{1,3}}      {{1},{2,3},{2,4}}
                      {{1},{1,2},{1,3}}    {{1},{2},{3},{4}}
                      {{1},{1,2},{2,3}}    {{1,2},{1,3},{1,4}}
                      {{1,2},{1,3},{2,3}}  {{1,2},{1,3},{2,4}}
                                           {{1},{2},{3},{1,4}}
                                           {{1},{2},{1,3},{1,4}}
                                           {{1},{2},{1,3},{2,4}}
                                           {{1},{2},{1,3},{3,4}}
                                           {{1},{1,2},{1,3},{1,4}}
                                           {{1},{1,2},{1,3},{2,4}}
                                           {{1},{1,2},{2,3},{2,4}}
                                           {{1},{1,2},{2,3},{3,4}}
                                           {{1},{2,3},{2,4},{3,4}}
                                           {{1,2},{1,3},{1,4},{2,3}}
                                           {{1,2},{1,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..500

Without the choice condition we have A322700, labeled A322661.
Without loops we have A368834, covering case of A134964.
For exactly n edges we have A368984, labeled A333331 (maybe).
The labeled version is A369140, covering case of A368927.
The labeled complement is A369142, covering case of A369141.
This is the covering case of A369145.
The complement is counted by A369147, covering case of A369146.
The complement without loops is A369202, covering case of A140637.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, labeled A006125 (shifted left).
A006129 counts covering graphs, unlabeled A002494.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A129271 counts connected choosable simple graphs, unlabeled A005703.
A133686 counts choosable labeled graphs, covering A367869.
Cf. A000088, A000612, A006649, A014068, A055621, A137916, A137917, A368835, A369194, A369199.

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],{1,2}]], Union@@#==Range[n]&&Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]]],{n,0,4}]

First differences of A369145.

Euler transform of A369289 with A369289(1) = 1. - Andrew Howroyd, Feb 02 2024

a(7) onwards from Andrew Howroyd, Feb 02 2024

A300626 Number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions.

Original entry on oeis.org

1, 1, 3, 11, 43, 187, 872, 4375, 23258, 130485, 767348, 4710715, 30070205, 198983975, 1361361925, 9607908808, 69812787049, 521377973359, 3996036977270, 31389624598631, 252408597286705, 2075472033455894, 17434190966525003, 149476993511444023, 1307022313790487959

Offset: 0

Gus Wiseman, Aug 17 2018

nonn

A free pure symmetric multifunction (with empty expressions allowed) f in EOME is either (case 1) a positive integer, or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where k >= 0, h is in EOME, each of the g_i for i = 1, ..., k is in EOME, and for i < j we have g_i <= g_j under a canonical total ordering of EOME, such as the Mathematica ordering of expressions.

Also the number of inequivalent colorings of orderless Mathematica expressions with n positions.

			Inequivalent representatives of the a(3) = 11 colorings:
  1[1,1]  1[2,2]  1[1,2]  1[2,3]
  1[1[]]  1[2[]]
  1[][1]  1[][2]
  1[1][]  1[2][]
  1[][][]

Row sums of A304485.

Cf. A000612, A007716, A052893, A053492, A277996, A279944, A280000, A317652, A317655, A317656, A317676.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + x*p*sExp(p)); p}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 30 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 30 2020

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

A330282 Number of fully chiral set-systems on n vertices.

Original entry on oeis.org

1, 2, 5, 52, 21521

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.

			The a(0) = 1 through a(2) = 5 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{1,2}}
             {{2},{1,2}}

Costrict (or T_0) set-systems are A326940.
The covering case is A330229.
The unlabeled version is A330294, with covering case A330295.
Achiral set-systems are A083323.
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, A319637, A330098, A330231, A330232, A330234.

graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Length[graprms[#]]==Length[Union@@#]!&]],{n,0,3}]

Binomial transform of A330229.

A369201 Number of unlabeled simple graphs with n vertices and n edges 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, 0, 1, 7, 30, 124, 507, 2036, 8216, 33515, 138557, 583040, 2503093, 10985364, 49361893, 227342301, 1073896332, 5204340846, 25874724616, 131937166616, 689653979583, 3693193801069, 20247844510508, 113564665880028, 651138092719098, 3813739129140469

Offset: 0

Gus Wiseman, Jan 22 2024

nonn

These are graphs with n vertices and n edges having at least two cycles in the same component.

			The a(0) = 0 through a(6) = 7 simple graphs:
  .  .  .  .  .  {{12}{13}{14}{23}{24}}  {{12}{13}{14}{15}{23}{24}}
                                         {{12}{13}{14}{15}{23}{45}}
                                         {{12}{13}{14}{23}{24}{34}}
                                         {{12}{13}{14}{23}{24}{35}}
                                         {{12}{13}{14}{23}{24}{56}}
                                         {{12}{13}{14}{23}{25}{45}}
                                         {{12}{13}{14}{25}{35}{45}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(6) = 7 unlabeled non-choosable graphs with 6 vertices and 6 edges.

Without the choice condition we have A001434, covering A006649.
The labeled version without choice is A116508, covering A367863, A367862.
The complement is counted by A137917, labeled A137916.
For any number of edges we have A140637, complement A134964.
For labeled set-systems we have A368600.
The case with loops is A368835, labeled A368596.
The labeled version is A369143, covering A369144.
A006129 counts covering graphs, unlabeled A002494.
A007716 counts unlabeled multiset partitions, connected A007718.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable simple graphs, unlabeled A005703.
Cf. A000088, A000612, A014068, A053530, A133686, A140638, A368601, 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}],{n}],Select[Tuples[#],UnsameQ@@#&]=={}&]]],{n,0,5}]

a(n) = A001434(n) - A137917(n).

a(25) onwards from Andrew Howroyd, Feb 02 2024

A002857 Number of Post functions of n variables.

Original entry on oeis.org

1, 3, 20, 996, 9333312, 6406603084568576, 16879085743296493582043922521915392, 717956902513121252476003434439730211917452457474409186632352788205535232

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Roger F. Wheeler, Complete propositional connectives. Z. Math. Logik Grundlagen Math. 7, 1961, 185-198.

Alois P. Heinz, Table of n, a(n) for n = 1..12
Jürgen Heller, Identifiability in probabilistic knowledge structures, J. Math. Psychol. 77, 46-57 (2017).
R. F. Wheeler, Complete propositional connectives, Z. Math. Logik Grundlagen Math. 7 1961 185-198. [Annotated scanned copy]
R. F. Wheeler, An asymptotic formula for the number of complete propositional connectives, Z. Math. Logik Grundlagen Math. 8 (1962), 1-4. [Annotated scanned copy]

Equals A000612/2 and A003180/4.

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, [])/4:
seq(a(n), n=1..8);  # Alois P. Heinz, Aug 14 2019

b[n_, i_, l_] := If[n==0, 2^Function[w, Sum[Product[2^GCD[t, l[[h]]], {h, 1, Length[l]}], {t, 1, w}]/w][LCM @@ l], If[i < 1, 0, Sum[b[n - i j, i-1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]]];
a[n_] := b[n, n, {}]/4;
Array[a, 8] (* Jean-François Alcover, Oct 27 2020, after Alois P. Heinz *)

Conjecture: a(n) = A055621(n) - A055152(n). - R. J. Mathar, Oct 14 2022

More terms from Vladeta Jovovic, Feb 23 2000

A318131 Number of non-isomorphic sets of finite (possibly empty) sets with union {1,2,...,n} and intersection {}.

Original entry on oeis.org

1, 1, 6, 60, 3836, 37325360, 25626412263611792, 67516342973185974276922865448446208, 2871827610052485009904013737758920847534777143951264797898686184985092096

Offset: 0

Gus Wiseman, Aug 18 2018

nonn

			Non-isomorphic representatives of the a(2) = 6 sets of sets:
  {{1},{2}}
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{1},{2},{1,2}}
  {{},{1},{2},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..12

Cf. A000371, A000612, A003465, A055621, A119563, A131288, A283877, A293606, A304997.

Cf. A318128, A318129, A318130, A318132.

sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Subsets[Range[n]]],And[Union@@#===Range[n],Intersection@@#=={}]&]]],{n,4}]

a(n) = 2*(A055621(n) - A055621(n-1)) = 2*(A000612(n) - 2*A000612(n-1) + A000612(n-2)) for n >= 2. - Andrew Howroyd, Jan 29 2024

a(5) onwards from Andrew Howroyd, Jan 29 2024

A318132 Number of non-isomorphic set-systems (finite sets of finite nonempty sets) with union {1,2,...,n} and intersection {}.

Original entry on oeis.org

1, 0, 2, 26, 1884, 18660728, 12813206113141264, 33758171486592987125648226573752576, 1435913805026242504952006868879460423733630400489039411798068453617852416

Offset: 0

Gus Wiseman, Aug 18 2018

nonn

			Non-isomorphic representatives of the a(3) = 26 set-systems:
  {{1},{2,3}}
  {{1},{2},{3}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,2},{1,3}}
  {{1},{2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,2,3}}
  {{1},{2},{1,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3}}
  {{1},{1,2},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..12

Cf. A000371, A000612, A003465, A055621, A119563, A131288, A283877, A293606, A304997.

Cf. A318128, A318129, A318130, A318131.

sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Rest[Subsets[Range[n]]]],And[Union@@#===Range[n],Intersection@@#=={}]&]]],{n,4}]

a(n) = A055621(n) - 2*A055621(n-1) = A000612(n) - 3*A000612(n-1) + 2*A000612(n-2) for n >= 2. - Andrew Howroyd, Jan 29 2024

a(5) onwards from Andrew Howroyd, Jan 29 2024

cp's OEIS Frontend

A368836 Triangle read by rows where T(n,k) is the number of unlabeled loop-graphs on up to n vertices with k loops and n-k non-loops.

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A369200 Number of unlabeled loop-graphs covering n vertices such that it is possible to choose a different vertex from each edge (choosable).

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A300626 Number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions.

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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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A330282 Number of fully chiral set-systems on n vertices.

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A369201 Number of unlabeled simple graphs with n vertices and n edges such that it is not possible to choose a different vertex from each edge (non-choosable).

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