A321405 -id:A321405 - cp's OEIS frontend

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

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.

A330124 Number of unlabeled set-systems with n vertices and no endpoints.

Original entry on oeis.org

1, 1, 2, 22, 1776

Offset: 0

Gus Wiseman, Dec 05 2019

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

			Non-isomorphic representatives of the a(3) = 22 set-systems:
  0
  {1}{2}{12}
  {12}{13}{23}
  {1}{23}{123}
  {12}{13}{123}
  {1}{2}{13}{23}
  {1}{2}{3}{123}
  {1}{12}{13}{23}
  {1}{2}{13}{123}
  {1}{12}{13}{123}
  {1}{12}{23}{123}
  {12}{13}{23}{123}
  {1}{2}{3}{12}{13}
  {1}{2}{12}{13}{23}
  {1}{2}{3}{12}{123}
  {1}{2}{12}{13}{123}
  {1}{2}{13}{23}{123}
  {1}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}
  {1}{2}{3}{12}{13}{123}
  {1}{2}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}{123}

Partial sums of the covering case A330196.
The labeled version is A330059.
The "multi" version is A302545.
Unlabeled set-systems with no endpoints counted by weight are A330054.
Unlabeled set-systems with no singletons are A317794.
Unlabeled set-systems counted by vertices are A000612.
Unlabeled set-systems counted by weight are A283877.
The case with no singletons is A320665.
Cf. A007716, A016031, A306005, A317795, A321405, A330052, A330055.

A368096 Triangle read by rows where T(n,k) is the number of non-isomorphic set-systems of length k and weight n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 5, 8, 3, 1, 0, 1, 8, 18, 13, 3, 1, 0, 1, 9, 32, 37, 15, 3, 1, 0, 1, 13, 55, 96, 59, 16, 3, 1, 0, 1, 14, 91, 209, 196, 74, 16, 3, 1, 0, 1, 19, 138, 449, 573, 313, 82, 16, 3, 1, 0, 1, 20, 206, 863, 1529, 1147, 403, 84, 16, 3, 1

Offset: 0

Gus Wiseman, Dec 28 2023

A set-system is a finite set of finite nonempty sets.

Conjecture: Column k = 2 is A101881.

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   2   1
   0   1   4   3   1
   0   1   5   8   3   1
   0   1   8  18  13   3   1
   0   1   9  32  37  15   3   1
   0   1  13  55  96  59  16   3   1
   0   1  14  91 209 196  74  16   3   1
   0   1  19 138 449 573 313  82  16   3   1
   ...
Non-isomorphic representatives of the set-systems counted in row n = 5:
  .  {12345}  {1}{1234}  {1}{2}{123}  {1}{2}{3}{12}  {1}{2}{3}{4}{5}
              {1}{2345}  {1}{2}{134}  {1}{2}{3}{14}
              {12}{123}  {1}{2}{345}  {1}{2}{3}{45}
              {12}{134}  {1}{12}{13}
              {12}{345}  {1}{12}{23}
                         {1}{12}{34}
                         {1}{23}{24}
                         {1}{23}{45}

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

Row sums are A283877, connected case A300913.
For multiset partitions we have A317533.
Counting connected components instead of edges gives A321194.
For set multipartitions we have A334550.
For strict multiset partitions we have A368099.
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A049311 counts non-isomorphic set multipartitions, connected A056156.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A316980 counts non-isomorphic strict multiset partitions, connected A319557.
A319559 counts non-isomorphic T_0 set-systems, connected A319566.
Cf. A101881, A255903, A302545, A306005, A317532, A317794, A317795, A319560, A321405, A368094, A368095.

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]}]]];
Table[Length[Union[brute /@ Select[mpm[n],UnsameQ@@#&&And@@UnsameQ@@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
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}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
T(n)={[Vecrev(p) | p <- Vec(G(n))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

Terms a(66) and beyond from Andrew Howroyd, Jan 11 2024

A368726 Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.

Original entry on oeis.org

1, 1, 3, 3, 8, 10, 26, 38, 93, 161, 381, 732, 1721, 3566, 8369, 18316, 43280, 98401, 234959, 549628, 1327726, 3175670, 7763500, 18905703, 46762513, 115613599, 289185492, 724438500, 1831398264, 4641907993, 11853385002, 30365353560

Offset: 0

Gus Wiseman, Jan 06 2024

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 10 multiset partitions:
  {{1}}  {{1,1}}    {{1},{1,1}}    {{1,1},{1,1}}      {{1},{1,1},{1,1}}
         {{1,2}}    {{2},{1,2}}    {{1,2},{1,2}}      {{1},{1,2},{2,2}}
         {{1},{1}}  {{1},{1},{1}}  {{1,2},{2,2}}      {{2},{1,2},{1,2}}
                                   {{1,3},{2,3}}      {{2},{1,2},{2,2}}
                                   {{1},{1},{1,1}}    {{2},{1,3},{2,3}}
                                   {{1},{2},{1,2}}    {{3},{1,3},{2,3}}
                                   {{2},{2},{1,2}}    {{1},{1},{1},{1,1}}
                                   {{1},{1},{1},{1}}  {{1},{2},{2},{1,2}}
                                                      {{2},{2},{2},{1,2}}
                                                      {{1},{1},{1},{1},{1}}

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

For edges of any size we have A007718.
This is the connected case of A320663.
The case of singletons and strict pairs is A368727, Euler transform A339888.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368186, A368598, A368599, A368731.

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]}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n], Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

Inverse Euler transform of A320663.

A368727 Number of non-isomorphic connected multiset partitions of weight n into singletons or strict pairs.

Original entry on oeis.org

1, 1, 2, 2, 5, 6, 15, 21, 49, 82, 184, 341, 766, 1530, 3428, 7249, 16394, 36009, 82492, 186485, 433096, 1001495, 2358182, 5554644, 13255532, 31718030, 76656602, 185982207, 454889643, 1117496012, 2764222322, 6868902152, 17172601190

Offset: 0

Gus Wiseman, Jan 06 2024

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 multiset partitions:
  {1}  {12}    {2}{12}    {12}{12}      {2}{12}{12}      {12}{12}{12}
       {1}{1}  {1}{1}{1}  {13}{23}      {2}{13}{23}      {12}{13}{23}
                          {1}{2}{12}    {3}{13}{23}      {13}{23}{23}
                          {2}{2}{12}    {1}{2}{2}{12}    {13}{24}{34}
                          {1}{1}{1}{1}  {2}{2}{2}{12}    {14}{24}{34}
                                        {1}{1}{1}{1}{1}  {1}{2}{12}{12}
                                                         {1}{2}{13}{23}
                                                         {2}{2}{12}{12}
                                                         {2}{2}{13}{23}
                                                         {2}{3}{13}{23}
                                                         {3}{3}{13}{23}
                                                         {1}{1}{2}{2}{12}
                                                         {1}{2}{2}{2}{12}
                                                         {2}{2}{2}{2}{12}
                                                         {1}{1}{1}{1}{1}{1}

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

For edges of any size we have A056156, with loops A007718.
This is the connected case of A339888.
Allowing loops {x,x} gives A368726, Euler transform A320663.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368598, A368599, A368731.

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]}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List /@ c[[1]]],Union@@s[[c[[1]]]]]]]]];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],And@@UnsameQ@@@#&&Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

Inverse Euler transform of A339888.

cp's OEIS Frontend

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

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

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A330124 Number of unlabeled set-systems with n vertices and no endpoints.

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A368096 Triangle read by rows where T(n,k) is the number of non-isomorphic set-systems of length k and weight n.

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A368726 Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.

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A368727 Number of non-isomorphic connected multiset partitions of weight n into singletons or strict pairs.

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