A319557 -id:A319557 - 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

A316980 Number of non-isomorphic strict multiset partitions of weight n.

Original entry on oeis.org

1, 1, 3, 8, 23, 63, 197, 588, 1892, 6140, 20734, 71472, 254090, 923900, 3446572, 13149295, 51316445, 204556612, 832467052, 3455533022, 14621598811, 63023667027, 276559371189, 1234802595648, 5606647482646, 25875459311317, 121324797470067, 577692044073205

Offset: 0

Gus Wiseman, Jul 18 2018

nonn

Also the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations, with no equal rows (or alternatively, with no equal columns).

Also the number of non-isomorphic multiset partitions of weight n with no equivalent vertices. 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.

			Non-isomorphic representatives of the a(3) = 8 multiset partitions with no equivalent vertices (first column) and with no equal blocks (second column):
      (111) <-> (111)
      (122) <-> (1)(11)
    (1)(11) <-> (122)
    (1)(22) <-> (1)(22)
    (2)(12) <-> (2)(12)
  (1)(1)(1) <-> (123)
  (1)(2)(2) <-> (1)(23)
  (1)(2)(3) <-> (1)(2)(3)

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

Cf. A000009, A001055, A007716, A007717, A020555, A045778, A130091.
Cf. A316974, A316978, A316979, A316981, A316983.
Cf. A049311, A059201, A319558, A319560, A319567.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, 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)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(p=sum(t=1, n, subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*polcoef(exp(p-subst(p,x,x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 21 2023

Euler transform of A319557. - Gus Wiseman, Sep 23 2018

a(7)-a(10) from Gus Wiseman, Sep 23 2018

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

A319719 Number of non-isomorphic connected antichains of multisets of weight n.

Original entry on oeis.org

1, 1, 3, 4, 10, 14, 48, 95, 305, 822, 2615

Offset: 0

Gus Wiseman, Sep 26 2018

In an antichain, no part is a proper submultiset of any other. The weight of an antichain is the sum of sizes of its parts. Weight is generally not the same as number of vertices. Connected antichains are also called clutters.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 10 connected antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{1},{1}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,2,2}}
   {{1,2,3,3}}
   {{1,2,3,4}}
   {{1,1},{1,1}}
   {{1,2},{1,2}}
   {{1,2},{2,2}}
   {{1,3},{2,3}}
   {{1},{1},{1},{1}}

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A001055, A001970, A007716, A007718, A056156, A096827, A253249, A285573, A293994, A318099, A319557, A319616-A319646, A319721.

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.

A321155 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with density -1 <= k < n-2.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 6, 6, 4, 1, 10, 14, 11, 4, 1, 22, 38, 38, 20, 6, 1, 42, 94, 111, 72, 28, 6, 1, 94, 250, 348, 278, 138, 42, 8, 1, 203, 648, 1044, 992, 596, 226, 56, 8, 1, 470, 1728, 3192, 3538, 2536, 1192, 370, 76, 10, 1

Offset: 1

Gus Wiseman, Oct 29 2018

The density of a multiset partition of weight n with e parts and v vertices is n - e - v. The weight of a multiset partition is the sum of sizes of its parts.

			Triangle begins:
    1
    2    1
    3    2    1
    6    6    4    1
   10   14   11    4    1
   22   38   38   20    6    1
   42   94  111   72   28    6    1
   94  250  348  278  138   42    8    1
  203  648 1044  992  596  226   56    8    1
  470 1728 3192 3538 2536 1192  370   76   10    1
Non-isomorphic representatives of the connected multiset partitions counted in row 5:
{1,2,3,4,5}         {1,2,3,4,4}       {1,2,2,3,3}     {1,1,2,2,2}   {1,1,1,1,1}
{1,4},{2,3,4}       {1,2},{2,3,3}     {1,2,3,3,3}     {1,2,2,2,2}
{4},{1,2,3,4}       {1,3},{2,3,3}     {1,1},{1,2,2}   {1},{1,1,1,1}
{2},{1,3},{2,3}     {2},{1,2,3,3}     {1},{1,2,2,2}   {1,1},{1,1,1}
{2},{3},{1,2,3}     {2,3},{1,2,3}     {1,2},{1,2,2}
{3},{1,3},{2,3}     {3},{1,2,3,3}     {1,2},{2,2,2}
{3},{3},{1,2,3}     {3,3},{1,2,3}     {2},{1,1,2,2}
{1},{2},{2},{1,2}   {1},{1},{1,2,2}   {2},{1,2,2,2}
{2},{2},{2},{1,2}   {1},{1,2},{2,2}   {2,2},{1,2,2}
{1},{1},{1},{1},{1} {1},{2},{1,2,2}   {1},{1},{1,1,1}
                    {2},{1,2},{1,2}   {1},{1,1},{1,1}
                    {2},{1,2},{2,2}
                    {2},{2},{1,2,2}
                    {1},{1},{1},{1,1}

First column is A125702. Row sums are A007718.

Cf. A007716, A030019, A052888, A056156, A134954, A147878, A317631, A319557, A320921.

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.

cp's OEIS Frontend

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

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A316980 Number of non-isomorphic strict multiset partitions of weight 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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A319719 Number of non-isomorphic connected antichains of multisets 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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A321155 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with density -1 <= k < n-2.

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

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

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