A319565 -id:A319565 - cp's OEIS frontend

A007718 Number of independent polynomial invariants of matrix of order n.

1, 1, 3, 6, 17, 40, 125, 354, 1159, 3774, 13113, 46426, 171027, 644038, 2493848, 9867688, 39922991, 164747459, 693093407, 2968918400, 12940917244, 57353242370, 258306634422, 1181572250326, 5486982683756, 25856584485254

Offset: 0

Colin Mallows

nonn

Also the number of non-isomorphic connected multiset partitions of weight n. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices. - Gus Wiseman, Sep 23 2018

			From _Gus Wiseman_, Sep 24 2018: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(4) = 17 connected multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}
        {{1},{1}}   {{1,2,3}}      {{1,2,2,2}}
                   {{1},{1,1}}     {{1,2,3,3}}
                   {{2},{1,2}}     {{1,2,3,4}}
                  {{1},{1},{1}}   {{1},{1,1,1}}
                                  {{1},{1,2,2}}
                                  {{2},{1,2,2}}
                                  {{3},{1,2,3}}
                                  {{1,1},{1,1}}
                                  {{1,2},{1,2}}
                                  {{1,2},{2,2}}
                                  {{1,3},{2,3}}
                                 {{1},{1},{1,1}}
                                 {{1},{2},{1,2}}
                                 {{2},{2},{1,2}}
                                {{1},{1},{1},{1}}
(End)

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

Cf. A007716, A007718, A056156, A319557, A319565, A319566.

Inverse Euler transform of A007716.

a(7)-a(25) from Franklin T. Adams-Watters, Jun 21 2011

a(0)=1 prepended by Andrew Howroyd, Jan 15 2023

A056156 Number of connected bipartite graphs with n edges, no isolated vertices and a distinguished bipartite block, up to isomorphism.

Original entry on oeis.org

1, 2, 3, 7, 12, 32, 67, 181, 458, 1295, 3642, 10975, 33448, 106424, 345964, 1159489, 3975367, 13977808, 50238606, 184629655, 692757132, 2652892219, 10359676617, 41233344350, 167171988557, 690054189750, 2898637406813, 12385234548345

Offset: 1

Vladeta Jovovic, Jul 30 2000

nonn

EULERi transform of A049311.

Also the number of non-isomorphic connected set multipartitions (multisets of sets) of weight n. The weight of a set multipartition is the sum of sizes of its parts. Weight is generally not the same as number of vertices. - Gus Wiseman, Sep 23 2018

			From _Gus Wiseman_, Sep 24 2018: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(4) = 7 connected set multipartitions:
  {{1}}   {{1,2}}     {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}   {{2},{1,2}}    {{3},{1,2,3}}
                    {{1},{1},{1}}   {{1,2},{1,2}}
                                    {{1,3},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{2},{2},{1,2}}
                                  {{1},{1},{1},{1}}
(End)

Jean-François Alcover, Table of n, a(n) for n = 1..102
Daniel R. Herber, Enhancements to the perfect matching approach for graph enumeration-based engineering challenges, Proceedings of the ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE 2020).
N. J. A. Sloane, Transforms

Cf. A007716, A007718, A056156, A319557, A319565, A319566.

A049311 = Cases[Import["https://oeis.org/A049311/b049311.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
EulerInvTransform[A049311] (* Jean-François Alcover, Mar 16 2020 *)

More terms from Max Alekseyev, Jul 22 2009

A319559 Number of non-isomorphic T_0 set systems of weight n.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 16, 35, 82, 200, 517, 1373, 3867, 11216, 33910, 105950

Offset: 0

Gus Wiseman, Sep 23 2018

In a set system, two vertices are equivalent if in every block the presence of the first is equivalent to the presence of the second. The T_0 condition means that there are no equivalent vertices.

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) = 7 set systems:
1:        {{1}}
2:      {{1},{2}}
3:     {{2},{1,2}}
      {{1},{2},{3}}
4:    {{1,3},{2,3}}
     {{1},{2},{1,2}}
     {{1},{3},{2,3}}
    {{1},{2},{3},{4}}
5:  {{1},{2,4},{3,4}}
    {{2},{3},{1,2,3}}
    {{2},{1,3},{2,3}}
    {{3},{1,3},{2,3}}
   {{1},{2},{3},{2,3}}
   {{1},{2},{4},{3,4}}
  {{1},{2},{3},{4},{5}}

Cf. A007716, A007718, A049311, A053419, A056156, A059201, A283877, A305854, A306006, A316980, A317757.

Cf. A319557, A319558, A319560, A319564, A319565, A319566, A319567.

a(11)-a(15) from Bert Dobbelaere, May 04 2025

A319558 The squarefree dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted without multiplicity. Then a(n) is the number of non-isomorphic multiset partitions of weight n whose squarefree dual is strict (no repeated blocks).

Original entry on oeis.org

1, 1, 3, 7, 21, 55, 169, 496, 1582, 5080, 17073

Offset: 0

Gus Wiseman, Sep 23 2018

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, a(2) = 3, and a(3) = 7 multiset partitions:
1:    {{1}}
2:   {{1,1}}
    {{1},{1}}
    {{1},{2}}
3:  {{1,1,1}}
   {{1},{1,1}}
   {{1},{2,2}}
   {{2},{1,2}}
  {{1},{1},{1}}
  {{1},{2},{2}}
  {{1},{2},{3}}

Cf. A007716, A007718, A049311, A053419, A056156, A059201, A283877, A316980.

Cf. A319557, A319559, A319560, A319564, A319565, A319566, A319567.

A319557 Number of non-isomorphic strict connected multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 5, 12, 30, 91, 256, 823, 2656, 9103, 31876, 116113, 432824, 1659692, 6508521, 26112327, 106927561, 446654187, 1900858001, 8236367607, 36306790636, 162724173883, 741105774720, 3428164417401, 16099059101049, 76722208278328, 370903316203353, 1818316254655097

Offset: 0

Gus Wiseman, Sep 23 2018

nonn

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Also the number of non-isomorphic connected T_0 multiset partitions of weight n. In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second. The T_0 condition means that there are no equivalent vertices.

			Non-isomorphic representatives of the a(4) = 12 strict connected multiset partitions:
    {{1,1,1,1}}
    {{1,1,2,2}}
    {{1,2,2,2}}
    {{1,2,3,3}}
    {{1,2,3,4}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{2},{1,2,2}}
   {{3},{1,2,3}}
   {{1,2},{2,2}}
   {{1,3},{2,3}}
  {{1},{2},{1,2}}
Non-isomorphic representatives of the a(4) = 12 connected T_0 multiset partitions:
     {{1,1,1,1}}
     {{1,2,2,2}}
    {{1},{1,1,1}}
    {{1},{1,2,2}}
    {{2},{1,2,2}}
    {{1,1},{1,1}}
    {{1,2},{2,2}}
    {{1,3},{2,3}}
   {{1},{1},{1,1}}
   {{1},{2},{1,2}}
   {{2},{2},{1,2}}
  {{1},{1},{1},{1}}

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

Cf. A007716, A007718, A049311, A056156, A283877, A316980.

Cf. A319558, A319559, A319560, A319564, A319565, A319566, A319567.

Inverse Euler transform of A316980.

Terms a(11) and beyond from Andrew Howroyd, Jan 19 2023

A319560 Number of non-isomorphic strict T_0 multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 6, 15, 40, 121, 353, 1107, 3550, 11818

Offset: 0

Gus Wiseman, Sep 23 2018

In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second. The T_0 condition means that there are no equivalent vertices.

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(4) = 15 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1},{2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1},{1,1}}
   {{1},{2,2}}
   {{2},{1,2}}
   {{1},{2},{3}}
4: {{1,1,1,1}}
   {{1,2,2,2}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{1},{2,2,2}}
   {{1},{2,3,3}}
   {{2},{1,2,2}}
   {{1,1},{2,2}}
   {{1,2},{2,2}}
   {{1,3},{2,3}}
   {{1},{2},{1,2}}
   {{1},{2},{2,2}}
   {{1},{2},{3,3}}
   {{1},{3},{2,3}}
   {{1},{2},{3},{4}}

Cf. A007716, A007718, A049311, A053419, A056156, A059201, A283877, A316980.

Cf. A319557, A319558, A319559, A319564, A319565, A319566, A319567.

A319566 Number of non-isomorphic connected T_0 set systems of weight n.

Original entry on oeis.org

1, 1, 0, 1, 2, 3, 8, 17, 41, 103, 276

Offset: 0

Gus Wiseman, Sep 23 2018

In a set system, two vertices are equivalent if in every block the presence of the first is equivalent to the presence of the second. The T_0 condition means that there are no equivalent vertices.

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(6) = 8 set systems:
1:        {{1}}
3:     {{2},{1,2}}
4:    {{1,3},{2,3}}
     {{1},{2},{1,2}}
5:  {{2},{3},{1,2,3}}
    {{2},{1,3},{2,3}}
    {{3},{1,3},{2,3}}
6: {{3},{1,4},{2,3,4}}
   {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,3},{2,4},{3,4}}
   {{1,4},{2,4},{3,4}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}

Cf. A007716, A007718, A049311, A056156, A059201, A283877, A316980.

Cf. A319557, A319558, A319559, A319560, A319564, A319565, A319567.

A321585 Number of connected nonnegative integer matrices with sum of entries equal to n and no zero rows or columns.

Original entry on oeis.org

1, 1, 3, 11, 52, 312, 2290, 19920, 200522, 2293677, 29389005, 416998371, 6490825772, 109972169413, 2014696874717, 39684502845893, 836348775861331, 18777970539419957, 447471215460930665, 11279275874429302811, 299844572529989373703, 8383794111721619471384, 245956060268568277412668

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(3) = 11 matrices:
  [3] [2 1] [1 2] [1 1 1]
.
  [2] [1 1] [1 1] [1] [1 0] [0 1]
  [1] [1 0] [0 1] [2] [1 1] [1 1]
.
  [1]
  [1]
  [1]

Andrew Howroyd, Table of n, a(n) for n = 0..100

Cf. A007718, A056156, A120733, A319557, A319565, A319647, A319616-A319629, A321584.

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/@#],Length[csm[Map[Last,GatherBy[#,First],{2}]]]==1]&]],{n,5}] (* Mathematica 7.0+ *)

NonZeroCols(M)={my(C=Vec(M)); Mat(vector(#C, n,  sum(k=1, n, (-1)^(n-k)*binomial(n,k)*C[k])))}
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(M=matrix(n,n,i,j,sum(k=1, n, binomial(i*j+k-1,k)*x^k, O(x*x^n) ))); Vec(1 + vecsum(vecsum(Vec( ConnectedMats( NonZeroCols( NonZeroCols(M)~))))))} \\ Andrew Howroyd, Jan 17 2024

a(7) onwards from Andrew Howroyd, Jan 17 2024

A319629 Number of non-isomorphic connected weight-n antichains of distinct multisets whose dual is also an antichain of distinct multisets.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 9, 29, 66, 189

Offset: 0

Gus Wiseman, Sep 25 2018

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},{2,2}} 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) = 9 antichains:
1: {{1}}
2: {{1,1}}
3: {{1,1,1}}
4: {{1,1,1,1}}
5: {{1,1,1,1,1}}
   {{1,1},{1,2,2}}
6: {{1,1,1,1,1,1}}
   {{1,1},{1,2,2,2}}
   {{1,1,2},{1,2,2}}
   {{1,1,2},{2,2,2}}
   {{1,1,2},{2,3,3}}
   {{1,1},{1,2},{2,2}}
   {{1,2},{1,3},{2,3}}
7: {{1,1,1,1,1,1,1}}
   {{1,1},{1,2,2,2,2}}
   {{1,1,1},{1,2,2,2}}
   {{1,1,2},{1,2,2,2}}
   {{1,1,2},{2,2,2,2}}
   {{1,1,2},{2,3,3,3}}
   {{1,1},{1,2},{2,2,2}}
   {{1,1},{1,2},{2,3,3}}
   {{1,2},{1,3},{2,3,3}}

Cf. A006126, A007716, A007718, A056156, A059201, A283877, A316980, A316983, A318099, A319557, A319558, A319565, A319616-A319646.

Euler transform is A319644.

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

cp's OEIS Frontend

A007718 Number of independent polynomial invariants of matrix of order n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A056156 Number of connected bipartite graphs with n edges, no isolated vertices and a distinguished bipartite block, up to isomorphism.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A319559 Number of non-isomorphic T_0 set systems of weight n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

Views

Author

Keywords

Comments

Examples

Crossrefs

A319557 Number of non-isomorphic strict connected multiset partitions of weight n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A319560 Number of non-isomorphic strict T_0 multiset partitions of weight n.

Views

Author

Keywords

Comments

Examples

Crossrefs

A319566 Number of non-isomorphic connected T_0 set systems of weight n.

Views

Author

Keywords

Comments

Examples

Crossrefs

A321585 Number of connected nonnegative integer matrices with sum of entries equal to n and no zero rows or columns.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A319629 Number of non-isomorphic connected weight-n antichains of distinct multisets whose dual is also an antichain of distinct multisets.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula