A321405 -id:A321405 - cp's OEIS frontend

A135588 Number of symmetric (0,1)-matrices with exactly n entries equal to 1 and no zero rows or columns.

1, 1, 2, 6, 20, 74, 302, 1314, 6122, 29982, 154718, 831986, 4667070, 27118610, 163264862, 1013640242, 6488705638, 42687497378, 288492113950, 1998190669298, 14177192483742, 102856494496050, 762657487965086, 5771613810502002, 44555989658479726, 350503696871063138

Offset: 0

Vladeta Jovovic, Feb 25 2008, Mar 03 2008, Mar 04 2008

nonn

			From _Gus Wiseman_, Nov 14 2018: (Start)
The a(4) = 20 matrices:
  [11]
  [11]
.
  [110][101][100][100][011][010][010][001][001]
  [100][010][011][001][100][110][101][010][001]
  [001][100][010][011][100][001][010][101][110]
.
  [1000][1000][1000][1000][0100][0100][0010][0010][0001][0001]
  [0100][0100][0010][0001][1000][1000][0100][0001][0100][0010]
  [0010][0001][0100][0010][0010][0001][1000][1000][0010][0100]
  [0001][0010][0001][0100][0001][0010][0001][0100][1000][1000]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..721 (first 51 terms from Vincenzo Librandi)

Row sums of A135589.

Cf. A049311, A054976, A101370, A104601, A104602, A120733, A138178, A283877, A316983, A320796, A321401, A321405.

Table[Sum[SeriesCoefficient[(1+x)^k*(1+x^2)^(k*(k-1)/2)/2^(k+1),{x,0,n}],{k,0,Infinity}],{n,0,20}] (* Vaclav Kotesovec, Jul 02 2014 *)
Join[{1},  Table[Length[Select[Subsets[Tuples[Range[n], 2], {n}], And[Union[First/@#]==Range[Max@@First/@#], Union[Last/@#]==Range[Max@@Last/@#], Sort[Reverse/@#]==#]&]], {n, 5}]] (* Gus Wiseman, Nov 14 2018 *)

G.f.: Sum_{n>=0} (1+x)^n*(1+x^2)^binomial(n,2)/2^(n+1).

G.f.: Sum_{n>=0} (Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*(1+x)^k*(1+x^2)^binomial(k,2)).

A330055 Number of non-isomorphic set-systems of weight n with no singletons or endpoints.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 3, 5, 16, 24, 90, 179, 567, 1475, 4623, 13650, 44475, 144110, 492017, 1706956, 6124330, 22442687, 84406276, 324298231, 1273955153, 5106977701, 20885538133, 87046940269, 369534837538, 1596793560371, 7019424870960, 31374394197536, 142514998263015

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 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(7) = 1 through a(10) = 16 set-systems:
  {12}{13}{123}  {12}{134}{234}    {12}{134}{1234}    {12}{1345}{2345}
                 {12}{34}{1234}    {123}{124}{134}    {123}{124}{1234}
                 {12}{13}{24}{34}  {12}{13}{14}{234}  {123}{145}{2345}
                                   {12}{13}{23}{123}  {12}{345}{12345}
                                   {12}{13}{24}{134}  {12}{13}{124}{134}
                                                      {12}{13}{124}{234}
                                                      {12}{13}{14}{1234}
                                                      {12}{13}{24}{1234}
                                                      {12}{13}{245}{345}
                                                      {12}{13}{45}{2345}
                                                      {12}{34}{123}{124}
                                                      {12}{34}{125}{345}
                                                      {12}{34}{135}{245}
                                                      {13}{24}{123}{124}
                                                      {12}{13}{14}{23}{24}
                                                      {12}{13}{24}{35}{45}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Wikipedia, Degree (graph theory)

The labeled version is A330056.
The "multi" version is A320665.
Non-isomorphic set-systems with no singletons are A306005.
Non-isomorphic set-systems with no endpoints are A330054.
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, A321405, A330052, A330058.

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)={my(g=x*Ser(WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)))); (1-x)*g-subst(g,x,x^2)}
S(q, t, k)={(x-x^2)*sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(sum(t=1, n, subst(K(q, t, n\t)-S(q,t,n\t),x,x^t)/t )), n)); s/n!)} \\ Andrew Howroyd, Jan 27 2024

a(11) onwards from Andrew Howroyd, Jan 27 2024

A330102 BII-number of the VDD-normalization of the set-system with BII-number n.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 5, 7, 1, 3, 3, 11, 33, 19, 19, 15, 4, 5, 33, 19, 20, 21, 37, 23, 5, 7, 19, 15, 37, 23, 51, 31, 4, 33, 5, 19, 20, 37, 21, 23, 5, 19, 7, 15, 37, 51, 23, 31, 20, 37, 37, 51, 52, 53, 53, 55, 21, 23, 23, 31, 53, 55, 55, 63, 64, 65, 65, 67, 68, 69, 69

Offset: 0

Gus Wiseman, Dec 04 2019

nonn

First differs from A330101 at a(148) = 274, A330101(148) = 545, with corresponding set-systems 274: {{2},{1,3},{1,4}} and 545: {{1},{2,3},{2,4}}.
A set-system is a finite set of finite nonempty sets of positive integers.
We define the VDD (vertex-degrees decreasing) normalization of a set-system to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of sets is first by length and then lexicographically.
For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
  Brute-force:    2067: {{1},{2},{1,3},{3,4}}
  Lexicographic:   165: {{1},{4},{1,2},{2,3}}
  VDD:             525: {{1},{3},{1,2},{2,4}}
  MM:              270: {{2},{3},{1,2},{1,4}}
  BII:             150: {{2},{4},{1,2},{1,3}}
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			56 is the BII-number of {{3},{1,3},{2,3}}, which has VDD-normalization {{1},{1,2},{1,3}} with BII-number 21, so a(56) = 21.

Wikipedia, Idempotence

This sequence is idempotent and its image/fixed points are A330100.
Non-isomorphic multiset partitions are A007716.
Unlabeled spanning set-systems counted by vertices are A055621.
Unlabeled set-systems counted by weight are A283877.
Cf. A000120, A000612, A048793, A070939, A300913, A319559, A321405, A326031, A326754, A330061, A330101.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
fbi[q_]:=If[q=={},0,Total[2^q]/2];
sysnorm[m_]:=If[Union@@m!={}&&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[fbi[fbi/@sysnorm[bpe/@bpe[n]]],{n,0,100}]

A368410 Number of non-isomorphic connected set-systems of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 15, 32, 80, 198, 528

Offset: 0

Gus Wiseman, Dec 25 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 set-systems:
  {1}  {12}  {123}    {1234}    {12345}      {123456}
             {2}{12}  {13}{23}  {14}{234}    {125}{345}
                      {3}{123}  {23}{123}    {134}{234}
                                {4}{1234}    {15}{2345}
                                {2}{13}{23}  {34}{1234}
                                {2}{3}{123}  {5}{12345}
                                {3}{13}{23}  {1}{14}{234}
                                             {12}{13}{23}
                                             {1}{23}{123}
                                             {13}{24}{34}
                                             {14}{24}{34}
                                             {3}{14}{234}
                                             {3}{23}{123}
                                             {3}{4}{1234}
                                             {4}{14}{234}

Wikipedia, Axiom of choice.

For unlabeled graphs we have A005703, connected case of A134964.
For labeled graphs we have A129271, connected case of A133686.
The complement for labeled graphs is A140638, connected case of A367867.
The complement without connectedness is A367903, ranks A367907.
Without connectedness we have A368095, ranks A367906,
Complement with repeats: A368097, connected case of A368411, ranks A355529.
The complement is counted by A368409, connected case of A368094.
With repeats allowed: A368412, connected case of A368098, ranks A368100.
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A001055, A140637, A302545, A306005, A321194, A321405, A367902, A368414, A368422.

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]}]]];
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]]]]]]]]];
Table[Length[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]

A330056 Number of set-systems with n vertices and no singletons or endpoints.

Original entry on oeis.org

1, 1, 1, 6, 1724, 66963208, 144115175600855641, 1329227995784915809349010517957163445, 226156424291633194186662080095093568675422295082604716043360995547325655259

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) = 6 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 covering case is A330057.
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, A008299, A302545, A317533, A317794, A319559, A320665, A321405, A330052, A330058.

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

\\ Here AS2(n,k) is A008299 (associated Stirling of 2nd kind)
AS2(n, k) = {sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) )}
a(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)) )))} \\ Andrew Howroyd, Jan 16 2023

Binomial transform of A330057.

a(n) = Sum_{k=0..n} Sum_{j=0..floor(k/2)} Sum_{i=0..k-2*j} (-1)^k * binomial(n,k) * 2^(2^(n-k)-(n-k)-1) * binomial(k,i) * AS2(k-i, j) * (2^(n-k)-1)^i * 2^(j*(n-k)) where AS2(n,k) are the associated Stirling numbers of the 2nd kind (A008299). - Andrew Howroyd, Jan 16 2023

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

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

A321401 Number of non-isomorphic strict self-dual multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 29, 57, 117, 240, 498

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 (or columns) are all different.

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}}.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 14 multiset partitions:
  {{1}}  {{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},{1,2,2}}
                    {{2},{1,2}}    {{1},{2,2,2}}      {{1,1},{2,2,2}}
                    {{1},{2},{3}}  {{2},{1,2,2}}      {{1,2},{1,2,2}}
                                   {{1},{2},{3,3}}    {{1},{2,2,2,2}}
                                   {{1},{3},{2,3}}    {{2},{1,2,2,2}}
                                   {{1},{2},{3},{4}}  {{1},{2,2},{3,3}}
                                                      {{1},{2},{3,3,3}}
                                                      {{1},{3},{2,3,3}}
                                                      {{2},{1,2},{3,3}}
                                                      {{2},{1,3},{2,3}}
                                                      {{1},{2},{3},{4,4}}
                                                      {{1},{2},{4},{3,4}}
                                                      {{1},{2},{3},{4},{5}}

Cf. A000219, A007716, A045778, A059201, A316980, A316983, A319560, A319616.

Cf. A320796, A320797, A321402, A321405, A321406, A321407.

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.

A321403 Number of non-isomorphic self-dual set multipartitions (multisets of sets) of weight n.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 10, 17, 32, 56, 98, 177, 335, 620, 1164, 2231, 4349, 8511, 16870, 33844, 68746, 140894, 291698, 610051, 1288594, 2745916, 5903988, 12805313, 28010036, 61764992, 137281977, 307488896, 693912297, 1577386813, 3611241900, 8324940862, 19321470086

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

Also the number of symmetric (0,1)-matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns.
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(1) = 1 through a(7) = 17 set multipartitions:
  {{1}}  {{1},{2}}  {{2},{1,2}}    {{1,2},{1,2}}      {{1},{2,3},{2,3}}
                    {{1},{2},{3}}  {{1},{1},{2,3}}    {{2},{1,3},{2,3}}
                                   {{1},{3},{2,3}}    {{3},{3},{1,2,3}}
                                   {{1},{2},{3},{4}}  {{1},{2},{2},{3,4}}
                                                      {{1},{2},{4},{3,4}}
                                                      {{1},{2},{3},{4},{5}}
.
  {{1,2},{1,3},{2,3}}        {{1,3},{2,3},{1,2,3}}
  {{3},{2,3},{1,2,3}}        {{1},{1},{1,4},{2,3,4}}
  {{1},{1},{1},{2,3,4}}      {{1},{2,3},{2,4},{3,4}}
  {{1},{2},{3,4},{3,4}}      {{1},{4},{3,4},{2,3,4}}
  {{1},{3},{2,4},{3,4}}      {{2},{1,2},{3,4},{3,4}}
  {{1},{4},{4},{2,3,4}}      {{2},{1,3},{2,4},{3,4}}
  {{2},{4},{1,2},{3,4}}      {{3},{4},{1,4},{2,3,4}}
  {{1},{2},{3},{3},{4,5}}    {{4},{4},{4},{1,2,3,4}}
  {{1},{2},{3},{5},{4,5}}    {{1},{1},{5},{2,3},{4,5}}
  {{1},{2},{3},{4},{5},{6}}  {{1},{2},{2},{2},{3,4,5}}
                             {{1},{2},{3},{4,5},{4,5}}
                             {{1},{2},{4},{3,5},{4,5}}
                             {{1},{2},{5},{5},{3,4,5}}
                             {{1},{3},{5},{2,3},{4,5}}
                             {{1},{2},{3},{4},{4},{5,6}}
                             {{1},{2},{3},{4},{6},{5,6}}
                             {{1},{2},{3},{4},{5},{6},{7}}
Inequivalent representatives of the a(6) = 10 matrices:
  [0 0 1] [1 1 0]
  [0 1 1] [1 0 1]
  [1 1 1] [0 1 1]
.
  [1 0 0 0] [1 0 0 0] [1 0 0 0] [1 0 0 0] [0 1 0 0]
  [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] [0 0 0 1]
  [1 0 0 0] [0 0 1 1] [0 1 0 1] [0 0 0 1] [1 1 0 0]
  [0 1 1 1] [0 0 1 1] [0 0 1 1] [0 1 1 1] [0 0 1 1]
.
  [1 0 0 0 0] [1 0 0 0 0]
  [0 1 0 0 0] [0 1 0 0 0]
  [0 0 1 0 0] [0 0 1 0 0]
  [0 0 1 0 0] [0 0 0 0 1]
  [0 0 0 1 1] [0 0 0 1 1]
.
  [1 0 0 0 0 0]
  [0 1 0 0 0 0]
  [0 0 1 0 0 0]
  [0 0 0 1 0 0]
  [0 0 0 0 1 0]
  [0 0 0 0 0 1]

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

Cf. A007716, A049311, A135588, A135589, A138178, A283877, A316983, A319616.

Cf. A320796, A320797, A321403, A321404, A321405.

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}
c(p, k)={polcoef((prod(i=2, #p, prod(j=1, i-1, (1 + x^(2*lcm(p[i], p[j])) + O(x*x^k))^gcd(p[i], p[j]))) * prod(i=1, #p, my(t=p[i]); (1 + x^t + O(x*x^k))^(t%2)*(1 + x^(2*t) + O(x*x^k))^(t\2) )), k)}
a(n)={my(s=0); forpart(p=n, s+=permcount(p)*c(p, n)); s/n!} \\ Andrew Howroyd, May 31 2023

Terms a(11) and beyond from Andrew Howroyd, May 31 2023

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.

cp's OEIS Frontend

A135588 Number of symmetric (0,1)-matrices with exactly n entries equal to 1 and no zero rows or columns.

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A330055 Number of non-isomorphic set-systems of weight n with no singletons or endpoints.

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A330102 BII-number of the VDD-normalization of the set-system with BII-number n.

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A368410 Number of non-isomorphic connected set-systems of weight n satisfying a strict version of the axiom of choice.

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A330056 Number of set-systems with n vertices and no singletons or endpoints.

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

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A321401 Number of non-isomorphic strict self-dual multiset partitions of weight n.

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

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