A319559 -id:A319559 - cp's OEIS frontend

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

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

A321515 Number of nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows and columns.

Original entry on oeis.org

1, 1, 3, 19, 137, 1209, 12899, 160395, 2276229, 36323217, 643848837, 12551081501, 266868756473, 6146455542737, 152439235077709, 4050427673024753, 114791270281213209, 3456412742412516649, 110191808168628510207, 3708004806262196242699, 131339701217968663631857

Offset: 0

Gus Wiseman, Nov 13 2018

nonn

			The a(3) = 19 matrices:
  [3] [2 1] [1 2]
.
  [2] [2 0] [1 1] [1 1] [1] [1 0] [1 0] [0 2] [0 1] [0 1]
  [1] [0 1] [1 0] [0 1] [2] [1 1] [0 2] [1 0] [2 0] [1 1]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Andrew Howroyd, Table of n, a(n) for n = 0..40

Cf. A007716, A120733, A283877, A316980, A319559, A321446, A321586, A321588.

multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@prs2mat[#],UnsameQ@@Transpose[prs2mat[#]]]&]],{n,5}]

\\ Q(m,n,wf) defined in A321588.
seq(n)={my(R=vectorv(n,m,Q(m,n,w->1/(1 - y^w) + O(y*y^n)))); for(i=2, #R, R[i] -= i*R[i-1]); Vec(1 + vecsum(vecsum(R)))} \\ Andrew Howroyd, Jan 24 2024

a(7) onwards from Andrew Howroyd, Jan 20 2024

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

A326942 Number of unlabeled T_0 sets of subsets of {1..n} that cover all n vertices.

Original entry on oeis.org

2, 2, 6, 58, 3770

Offset: 0

Gus Wiseman, Aug 07 2019

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			Non-isomorphic representatives of the a(0) = 2 through a(2) = 6 sets of subsets:
  {}    {{1}}     {{1},{2}}
  {{}}  {{},{1}}  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A003181.
The case without empty edges is A319637.
The labeled version is A326939.
The non-covering version is A326949 (partial sums).
Cf. A000371, A003180, A055621, A059201, A316978, A319559, A319564, A326907, A326941, A326943, A326946.

a(n) = 2 * A319637(n).

A326959 Number of T_0 set-systems covering a subset of {1..n} that are closed under intersection.

Original entry on oeis.org

1, 2, 5, 22, 297, 20536, 16232437, 1063231148918, 225402337742595309857

Offset: 0

Gus Wiseman, Aug 13 2019

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 1 through a(3) = 22 set-systems:
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{2}}        {{2}}
             {{1},{1,2}}  {{3}}
             {{2},{1,2}}  {{1},{1,2}}
                          {{1},{1,3}}
                          {{2},{1,2}}
                          {{2},{2,3}}
                          {{3},{1,3}}
                          {{3},{2,3}}
                          {{1},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1},{1,2},{1,2,3}}
                          {{1},{1,3},{1,2,3}}
                          {{2},{1,2},{1,2,3}}
                          {{2},{2,3},{1,2,3}}
                          {{3},{1,3},{1,2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{1},{1,2},{1,3},{1,2,3}}
                          {{2},{1,2},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

The covering case is A309615.
T_0 set-systems are A326940.
The version with empty edges allowed is A326945.
Cf. A051185, A058891, A059201, A316978, A319559, A309615, A319637, A326943, A326944, A326946, A326947, A326959.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Binomial transform of A309615.

a(5)-a(8) from Andrew Howroyd, Aug 14 2019

A321588 Number of connected nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows and columns.

Original entry on oeis.org

1, 1, 1, 9, 29, 181, 1285, 10635, 102355, 1118021, 13637175, 184238115, 2727293893, 43920009785, 764389610843, 14297306352937, 286014489487815, 6093615729757841, 137750602009548533, 3293082026520294529, 83006675263513350581, 2200216851785981586729, 61180266502369886181253

Offset: 0

Gus Wiseman, Nov 13 2018

nonn

A matrix is connected if the positions in each row (or each column) of the nonzero entries form a connected hypergraph.

			The a(4) = 29 matrices:
4 31 13
.
3 21 21 20 12 12 11 110 11 110 101 101 1 10 10 02 011 011 01 01
1 10 01 11 10 01 20 101 02 011 110 011 3 21 12 11 110 101 21 12
.
11 11 10 10 01 01
10 01 11 01 11 10
01 10 01 11 10 11

Andrew Howroyd, Table of n, a(n) for n = 0..40

Cf. A007718, A056156, A059201, A120733, A283877, A316980, A319557, A319558, A319559, A319565, A319647, A319616-A319629, A321446, A321515, A369285.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@prs2mat[#],UnsameQ@@Transpose[prs2mat[#]],Length[csm[Map[Last,GatherBy[#,First],{2}]]]==1]&]],{n,6}]

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,wf)={prod(j=1, #q, wf(t*q[j]))-1}
Q(m,n,wf=w->2)={my(s=0); forpart(p=m, s+=(-1)^#p*permcount(p)*exp(-sum(t=1, n, (-1)^t*x^t*K(p,t,wf)/t, O(x*x^n))) ); Vec((-1)^m*serchop(serlaplace(s),1), -n)}
ConnectedMats(M)={my([m, n]=matsize(M), R=matrix(m, n)); for(m=1, m, for(n=1, n, R[m, n] = M[m, n] - sum(i=1, m-1, sum(j=1, n-1, binomial(m-1, i-1)*binomial(n, j)*R[i, j]*M[m-i, n-j])))); R}
seq(n)={my(R=vectorv(n,m,Q(m,n,w->1/(1 - y^w) + O(y*y^n)))); for(i=2, #R, R[i] -= i*R[i-1]); Vec(1 + vecsum( vecsum( Vec( ConnectedMats( Mat(R))))))} \\ Andrew Howroyd, Jan 24 2024

a(7) onwards from Andrew Howroyd, Jan 24 2024

A326948 Number of connected T_0 set-systems on n vertices.

Original entry on oeis.org

1, 1, 3, 86, 31302, 2146841520, 9223371978880250448, 170141183460469231408869283342774399392, 57896044618658097711785492504343953919148780260559635830120038252613826101856

Offset: 0

Gus Wiseman, Aug 08 2019

nonn

The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(3) = 86 set-systems:
  {12}{13}         {1}{2}{13}{123}     {1}{2}{3}{13}{23}
  {12}{23}         {1}{2}{23}{123}     {1}{2}{3}{13}{123}
  {13}{23}         {1}{3}{12}{13}      {1}{2}{3}{23}{123}
  {1}{2}{123}      {1}{3}{12}{23}      {1}{2}{12}{13}{23}
  {1}{3}{123}      {1}{3}{12}{123}     {1}{2}{12}{13}{123}
  {1}{12}{13}      {1}{3}{13}{23}      {1}{2}{12}{23}{123}
  {1}{12}{23}      {1}{3}{13}{123}     {1}{2}{13}{23}{123}
  {1}{12}{123}     {1}{3}{23}{123}     {1}{3}{12}{13}{23}
  {1}{13}{23}      {1}{12}{13}{23}     {1}{3}{12}{13}{123}
  {1}{13}{123}     {1}{12}{13}{123}    {1}{3}{12}{23}{123}
  {2}{3}{123}      {1}{12}{23}{123}    {1}{3}{13}{23}{123}
  {2}{12}{13}      {1}{13}{23}{123}    {1}{12}{13}{23}{123}
  {2}{12}{23}      {2}{3}{12}{13}      {2}{3}{12}{13}{23}
  {2}{12}{123}     {2}{3}{12}{23}      {2}{3}{12}{13}{123}
  {2}{13}{23}      {2}{3}{12}{123}     {2}{3}{12}{23}{123}
  {2}{23}{123}     {2}{3}{13}{23}      {2}{3}{13}{23}{123}
  {3}{12}{13}      {2}{3}{13}{123}     {2}{12}{13}{23}{123}
  {3}{12}{23}      {2}{3}{23}{123}     {3}{12}{13}{23}{123}
  {3}{13}{23}      {2}{12}{13}{23}     {1}{2}{3}{12}{13}{23}
  {3}{13}{123}     {2}{12}{13}{123}    {1}{2}{3}{12}{13}{123}
  {3}{23}{123}     {2}{12}{23}{123}    {1}{2}{3}{12}{23}{123}
  {12}{13}{23}     {2}{13}{23}{123}    {1}{2}{3}{13}{23}{123}
  {12}{13}{123}    {3}{12}{13}{23}     {1}{2}{12}{13}{23}{123}
  {12}{23}{123}    {3}{12}{13}{123}    {1}{3}{12}{13}{23}{123}
  {13}{23}{123}    {3}{12}{23}{123}    {2}{3}{12}{13}{23}{123}
  {1}{2}{3}{123}   {3}{13}{23}{123}    {1}{2}{3}{12}{13}{23}{123}
  {1}{2}{12}{13}   {12}{13}{23}{123}
  {1}{2}{12}{23}   {1}{2}{3}{12}{13}
  {1}{2}{12}{123}  {1}{2}{3}{12}{23}
  {1}{2}{13}{23}   {1}{2}{3}{12}{123}

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

The same with covering instead of connected is A059201, with unlabeled version A319637.
The non-T_0 version is A323818 (covering) or A326951 (not-covering).
The non-connected version is A326940, with unlabeled version A326946.
Cf. A000371, A003465, A245567, A316978, A319559, A319564, A326939, A326941, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&Length[csm[#]]<=1&&UnsameQ@@dual[#]&]],{n,0,3}]

Logarithmic transform of A059201.

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

A330195 BII-number of the BII-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, 12, 13, 13, 15, 4, 5, 12, 13, 20, 21, 22, 23, 5, 7, 13, 15, 22, 23, 30, 31, 4, 12, 5, 13, 20, 22, 21, 23, 5, 13, 7, 15, 22, 30, 23, 31, 20, 22, 22, 30, 52, 53, 53, 55, 21, 23, 23, 31, 53, 55, 55, 63, 64, 65, 65, 67, 68, 69

Offset: 0

Gus Wiseman, Dec 05 2019

nonn

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.
We define the BII-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, and finally taking the representative with the smallest BII-number.
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}}

Wikipedia, Idempotence

This sequence is idempotent and its image/fixed points are A330109.
A subset of A326754.
Unlabeled spanning set-systems counted by vertices are A055621.
Unlabeled set-systems counted by weight are A283877.
BII-weight is A326031.
Cf. A000120, A000612, A007716, A048793, A070939, A319559, A330061, A330101, A330102, A330194.
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];
biinorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],biinorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],fbi[fbi/@#]&]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Table[fbi[fbi/@biinorm[bpe/@bpe[n]]],{n,0,100}]

a(n) <= n.

cp's OEIS Frontend

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

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A321515 Number of nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows and columns.

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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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A326942 Number of unlabeled T_0 sets of subsets of {1..n} that cover all n vertices.

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A326959 Number of T_0 set-systems covering a subset of {1..n} that are closed under intersection.

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A321588 Number of connected nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows and columns.

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