A319566 -id:A319566 - 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

A319565 Number of non-isomorphic connected strict T_0 multiset partitions of weight n.

Original entry on oeis.org

1, 1, 1, 4, 8, 21, 62, 175, 553, 1775, 6007

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) = 8 multiset partitions:
1:      {{1}}
2:     {{1,1}}
3:    {{1,1,1}}
      {{1,2,2}}
     {{1},{1,1}}
     {{2},{1,2}}
4:   {{1,1,1,1}}
     {{1,2,2,2}}
    {{1},{1,1,1}}
    {{1},{1,2,2}}
    {{2},{1,2,2}}
    {{1,2},{2,2}}
    {{1,3},{2,3}}
   {{1},{2},{1,2}}

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

Cf. A319557, A319558, A319559, A319560, A319564, A319566, A319567.

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.

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

cp's OEIS Frontend

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

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

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

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

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A319565 Number of non-isomorphic connected strict T_0 multiset partitions of weight n.

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

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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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