A049312 -id:A049312 - cp's OEIS frontend

A049311 Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations.

1, 3, 6, 16, 34, 90, 211, 558, 1430, 3908, 10725, 30825, 90156, 273234, 848355, 2714399, 8909057, 30042866, 103859678, 368075596, 1335537312, 4958599228, 18820993913, 72980867400, 288885080660, 1166541823566, 4802259167367, 20141650236664

Offset: 1

Peter J. Cameron

Also the number of bipartite graphs with n edges, no isolated vertices and a distinguished bipartite block, up to isomorphism.
The EULERi transform (A056156) is also interesting.
a(n) is also the number of non-isomorphic set multipartitions (multisets of sets) of weight n. - Gus Wiseman, Mar 17 2017

			E.g. a(2) = 3: two ones in same row, two ones in same column, or neither.
a(3) = 6 is coefficient of x^3 in (1/36)*((1 + x)^9 + 6*(1 + x)^3*(1 + x^2)^3 + 8*(1 + x^3)^3 + 9*(1 + x)*(1 + x^2)^4 + 12*(1 + x^3)*(1 + x^6))=1 + x + 3*x^2 + 6*x^3 + 7*x^4 + 7*x^5 + 6*x^6 + 3*x^7 + x^8 + x^9.
There are a(3) = 6 binary matrices with 3 ones, with no zero rows or columns, up to row and column permutation:
  [1 0 0] [1 1 0] [1 0] [1 1] [1 1 1] [1]
  [0 1 0] [0 0 1] [1 0] [1 0] ....... [1].
  [0 0 1] ....... [0 1] ............. [1]
Non-isomorphic representatives of the a(3)=6 set multipartitions are: ((123)), ((1)(23)), ((2)(12)), ((1)(1)(1)), ((1)(2)(2)), ((1)(2)(3)). - _Gus Wiseman_, Mar 17 2017

Aliaksandr Siarhei, Table of n, a(n) for n = 1..102
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394.
Peter J. Cameron, Problems on Permutation Groups, see Problem 3
Index entries for sequences related to binary matrices.

Main diagonal of A321609.
Cf. A049312, A048194, A028657, A055192, A055599, A052371, A052370, A053304, A053305, A007716, A002724.
Cf. A057149, A057150, A057151, A057152.
Cf. A034691, A056156, A089259, A116540, A283877.

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, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 16 2023

Calculate number of connected bipartite graphs + number of connected bipartite graphs with no duality automorphism, then apply EULER transform.

a(n) is the coefficient of x^n in the cycle index Z(S_n X S_n; 1+x, 1+x^2, ...), where S_n X S_n is Cartesian product of symmetric groups S_n of degree n.

More terms and formula from Vladeta Jovovic, Jul 29 2000
a(19)-a(28) from Max Alekseyev, Jul 22 2009
a(29)-a(102) from Aliaksandr Siarhei, Dec 13 2013
Name edited by Gus Wiseman, Dec 18 2018

A028657 Triangle read by rows: T(n,k) = number of n-node graphs with k nodes in distinguished bipartite block, k = 0..n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 22, 36, 22, 6, 1, 1, 7, 34, 87, 87, 34, 7, 1, 1, 8, 50, 190, 317, 190, 50, 8, 1, 1, 9, 70, 386, 1053, 1053, 386, 70, 9, 1, 1, 10, 95, 734, 3250, 5624, 3250, 734, 95, 10, 1, 1, 11, 125, 1324, 9343, 28576, 28576, 9343, 1324, 125, 11, 1

Offset: 0

Vladeta Jovovic, Jun 16 2000

Also, row n gives the number of unlabeled bicolored graphs having k nodes of one color and n-k nodes of the other color; the color classes are not interchangeable.
Also the number of principal transversal matroids (also known as fundamental transversal matroids) of size n and rank k (originally enumerated by Brylawski). - Gordon F. Royle, Oct 30 2007
This sequence is also obtained if we read the array A(m,n) = number of inequivalent m X n binary matrices by antidiagonals, where equivalence means permutations of rows or columns (m>=0, n>=0) [Kerber]. - N. J. A. Sloane, Sep 01 2013

			The triangle T(n,k) begins:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  7,  4,  1;
  1,  5, 13, 13,  5,  1;
  1,  6, 22, 36, 22,  6,  1;
  ...
For example, there are 36 graphs on 6 nodes with a distinguished bipartite block with 3 nodes.
The array A(m,n) (m>=0, n>=0) (see Comments) begins:
  1 1  1    1     1      1        1         1           1 ...
  1 2  3    4     5      6        7         8           9 ...
  1 3  7   13    22     34       50        70          95 ...
  1 4 13   36    87    190      386       734        1324 ...
  1 5 22   87   317   1053     3250      9343       25207 ...
  1 6 34  190  1053   5624    28576    136758      613894 ...
  1 7 50  386  3250  28576   251610   2141733    17256831 ...
  1 8 70  734  9343 136758  2141733  33642660   508147108 ...
  1 9 95 1324 25207 613894 17256831 508147108 14685630688 ...
... - _N. J. A. Sloane_, Sep 01 2013

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

Alois P. Heinz, Rows n = 0..45, flattened (first 20 rows from R. W. Robinson)
Thomas H. Brylawski, An affine representation for transversal geometries, Studies in Appl. Math. 54 (1975), no. 2, 143-160.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43. See Table 1.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy] See Table 1.
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
Vladeta Jovovic, Binary matrices up to row and column permutations
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
Carlos Simpson, Learning proofs for the classification of nilpotent semigroups, arXiv:2106.03015 [cs.LG], 2021.
Index to OEIS entries related to number of inequivalent matrices modulo permutation of row and columns.

Row sums give A049312.
A246106 is a very similar array.
Cf. A055080, A049312, A052265, A242093.
Diagonals of the array A(m,n) give A002724, A002725, A002728.
Rows (or columns) give A002623, A002727, A006148, A052264.
A(n,k) = A353585(2, n, k).

b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
    end:
g:= proc(n, k) option remember; add(add(2^add(add(igcd(i, j)*
      coeff(s, x, i)* coeff(t, x, j), j=1..degree(t)),
      i=1..degree(s))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,
      i=1..degree(s))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,
      i=1..degree(t)), t=b(n+k$2)), s=b(n$2))
    end:
A:= (n, k)-> g(min(n, k), abs(n-k)):
seq(seq(A(n, d-n), n=0..d), d=0..14); # Alois P. Heinz, Aug 01 2014

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Union[ Flatten[ Table[ Function[ {p}, p + j*x^i] /@ b[n - i*j, i-1], {j, 0, n/i}]]]]];
g[n_, k_] := g[n, k] = Sum[ Sum[ 2^Sum[ Sum[GCD[i, j] * Coefficient[s, x, i] * Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}] / Product[i^Coefficient[s, x, i] * Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}] / Product[i^Coefficient[t, x, i] * Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n+k, n+k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n-k]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

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)={sum(j=1, #q, gcd(t, q[j]))}
A(n, m)={my(s=0); forpart(q=m, s+=permcount(q)*polcoef(exp(sum(t=1, n, 2^K(q, t)/t*x^t) + O(x*x^n)), n)); s/m!}
{ for(r=0, 10, for(k=0, r, print1(A(r-k,k), ", ")); print) } \\ Andrew Howroyd, Mar 25 2020

\\ G(k,x) gives k-th column as rational function (see Jovovic link).
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}
Fix(q,x)={my(v=divisors(lcm(Vec(q))), u=apply(t->2^sum(j=1, #q, gcd(t, q[j])), v)); 1/prod(i=1, #v, my(t=v[i]); (1-x^t)^(sum(j=1, i, my(d=t/v[j]); if(!frac(d), moebius(d)*u[j]))/t))}
G(m,x)={my(s=0); forpart(q=m, s+=permcount(q)*Fix(q,x)); s/m!}
T(n,k)={my(m=max(k, n-k)); polcoef(G(n-m, x + O(x*x^m)), m)} \\ Andrew Howroyd, Mar 26 2020

A028657(n,k)=A353585(2, n, k) \\ M. F. Hasler, May 01 2022

A(m,n) = Sum_{p in P(m), q in P(n)} 2^Sum_{i in p, j in q} gcd(i,j) / (N(p) N(q)) where P(m) are the partition of m (see e.g., A036036), N(p) = Product_{distinct parts x in p} x^m(x)*m(x)!, m(x) = multiplicity of x in p. [corrected by Anders Kaseorg, Oct 04 2024]

A048194 Total number of split graphs (chordal + chordal complement) on n vertices.

Original entry on oeis.org

1, 2, 4, 9, 21, 56, 164, 557, 2223, 10766, 64956, 501696, 5067146, 67997750, 1224275498, 29733449510, 976520265678, 43425320764422, 2616632636247976, 213796933371366930, 23704270652844196754, 3569464106212250952762, 730647291666881838671052

Offset: 1

Gordon F. Royle

Also number of bipartite graphs with n vertices and no isolated vertices in distinguished bipartite block, up to isomorphism; so a(n) equals first differences of A049312. - Vladeta Jovovic, Jun 17 2000
All split graphs are perfect. - Falk Hüffner, Nov 29 2015
Inverse Euler transform gives A007776 with initial 1. - Andrew Howroyd, Oct 03 2018

Alois P. Heinz, Table of n, a(n) for n = 1..40
B. A. Chat, S. Pirzada, and A. Iványi, Recognition of split-graphic sequences, Acta Universitatis Sapientiae, Informatica, 6, 2 (2014) 252-286.
Karen L. Collins and Ann N. Trenk, Finding Balance: Split Graphs and Related Classes, arXiv:1706.03092 [math.CO], June 2017.
Karen L. Collins and Ann N. Trenk, Finding Balance: Split Graphs and Related Classes, Electron. J. Combin., 25 (2018), #P1.73.
S. Hougardy, Home Page.
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
Vladeta Jovovic, Binary matrices up to row and column permutations.
Gordon F. Royle, Counting set covers and split graphs, J. Integer Seqs., Vol. 3 (2000), #00.2.6.
J. M. Troyka, Split graphs: combinatorial species and asymptotics, Electron. J. Combin., 26 (2019), #P2.42.
J. M. Troyka, Split graphs: combinatorial species and asymptotics, arXiv:1803.07248 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Split Graph.
Index entries for sequences related to posets

Cf. A007776, A048192, A048193, A049312, A055080, A263859.

Detlef Pauly remarks that this is the unlabeled analog of A001831.

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[ Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
a[d_] := Sum[A[n, d - n], {n, 0, d}] - Sum[A[n, d - n - 1], {n, 0, d - 1}];
Table[a[n], {n, 1, 25}] (* Jean-François Alcover, May 26 2019, after Alois P. Heinz in A049312 *)

a(n) = A049312(n) - A049312(n-1) (see the Collins and Trenk link, Thms. 5 and 15). - Justin M. Troyka, Oct 29 2018
a(n) ~ A049312(n) ~ (1/n!) * Sum_{k=0..n} binomial(n,k) * 2^(k(n-k)) (see the Troyka link, Thms. 3.7 and 3.10). - Justin M. Troyka, Oct 29 2018
a(n) = A263859(n,1) + 1. - Geoffrey Critzer, Feb 05 2024

A007776 Number of connected posets with n elements of height 1.

Original entry on oeis.org

1, 2, 4, 10, 27, 88, 328, 1460, 7799, 51196, 422521, 4483460, 62330116, 1150504224, 28434624153, 945480850638, 42417674401330, 2572198227615998, 211135833162079184, 23487811567341121158, 3545543330739039981738, 727053904070651775719646

Offset: 2

Georg Wambach (gw(AT)informatik.Uni-Koeln.de)

nonn

Inverse Euler transform of A048194 and A049312. - Detlef Pauly (dettodet(AT)yahoo.de) and Vladeta Jovovic, Jul 25 2003
Essentially the same as A318870. - Georg Fischer, Oct 02 2018
Number of connected digraphs on n unlabeled nodes where every node has indegree 0 or outdegree 0 and there are no isolated nodes.  - Andrew Howroyd, Oct 03 2018

Andrew Howroyd, Table of n, a(n) for n = 2..50 (terms 2..40 from Alois P. Heinz)
N. J. A. Sloane, Transforms
J. Textor, A. Idelberger, and M. Liskiewicz, Learning from Pairwise Marginal Independencies, arXiv:1508.00280 [cs.AI], 2015.
Index entries for sequences related to posets

Cf. A005142, A002031 (labeled case), A048194, A049312, A055192, A318870, column 1 of A342500.

mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
b[d_] := Sum[A[n, d - n], {n, 0, d}];
EULERi[Array[b, 30]] // Rest (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz in A049312 *)

Inverse Euler transform of A055192. - Andrew Howroyd, Oct 03 2018

More terms from Vladeta Jovovic, Jul 25 2003

Offset corrected by Andrew Howroyd, Oct 03 2018

A055192 Number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block, up to isomorphism.

Original entry on oeis.org

1, 2, 5, 12, 35, 108, 393, 1666, 8543, 54190, 436740, 4565450, 62930604, 1156277748, 28509174012, 946786816168, 42448800498744, 2573207315483554, 211180300735118954, 23490473719472829824, 3545759835559406756008, 727077827560669587718290

Offset: 2

Vladeta Jovovic, Jun 18 2000

nonn

Also the number of connected split graphs on n vertices (cf. A048194). - Falk Hüffner, Dec 01 2015

Inverse Euler transform is A007776. - Andrew Howroyd, Oct 03 2018

Alois P. Heinz, Table of n, a(n) for n = 2..40

Equals second differences of A049312.
Cf. A007776, A024206, A055609, A055082, A055083, A055084.
Row sums of A056152 and also of A122083.

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[ Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
A049312[d_] := Sum[A[n, d - n], {n, 0, d}];
Differences[Table[A049312[n], {n, 0, 23}], 2] (* Jean-François Alcover, Sep 05 2019, after Alois P. Heinz in A049312 *)

A056152 Triangular array giving number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block with k=1..n-1 vertices, up to isomorphism.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 8, 17, 8, 1, 1, 11, 42, 42, 11, 1, 1, 15, 91, 179, 91, 15, 1, 1, 19, 180, 633, 633, 180, 19, 1, 1, 24, 328, 2001, 3835, 2001, 328, 24, 1, 1, 29, 565, 5745, 20755, 20755, 5745, 565, 29, 1, 1, 35, 930, 15274, 102089, 200082, 102089

Offset: 2

Vladeta Jovovic, Jul 29 2000

Also table read by rows: for 0 < k < n, a(n, k) = number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block with k vertices, up to isomorphism.
a(n, k) is the number of isomorphism classes of finite subdirectly irreducible almost distributive lattices in which the N-quotient has k upper covers and (n - k) lower covers. - David Wasserman, Feb 11 2002
Also, row n gives the number of unlabeled bicolored graphs having k nodes of one color and n-k nodes of the other color, with no isolated nodes; the color classes are not interchangeable.

			Triangle begins:
  1;
  1,  1;
  1,  3,   1;
  1,  5,   5,   1;
  1,  8,  17,   8,  1;
  1, 11,  42,  42,  11,  1;
  1, 15,  91, 179,  91,  15,  1;
  1, 19, 180, 633, 633, 180, 19, 1;
  ...
There are 17 bipartite graphs with 6 vertices, no isolated vertices and a distinguished bipartite block with 3 vertices, or equivalently, there are 17 3 X 3 binary matrices with no zero rows or columns, up to row and column permutation:
[0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1]
[0 0 1] [0 0 1] [0 1 0] [0 1 0] [0 1 0] [0 1 1] [0 1 1] [0 1 1] [1 1 0]
[1 1 0] [1 1 1] [1 0 0] [1 0 1] [1 1 1] [1 0 1] [1 1 0] [1 1 1] [1 1 0]
and
[0 0 1] [0 0 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [1 1 1]
[1 1 0] [1 1 1] [0 1 1] [0 1 1] [1 0 1] [1 0 1] [1 1 1] [1 1 1]
[1 1 1] [1 1 1] [1 0 1] [1 1 1] [1 1 0] [1 1 1] [1 1 1] [1 1 1].

J. G. Lee, Almost Distributive Lattice Varieties, Algebra Universalis, 21 (1985), 280-304.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43. See Table 2.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy] See Table 2.

Columns k=1..6 are A000012, A024206, A055609, A055082, A055083, A055084.
Row sums give A055192.
See A122083 for another version of this triangle.
Cf. A049312, A048194, A028657, A049311.

A339832 Number of bicolored graphs on n unlabeled nodes such that black nodes are not adjacent to each other.

Original entry on oeis.org

1, 2, 5, 14, 50, 230, 1543, 16252, 294007, 9598984, 577914329, 64384617634, 13264949930889, 5055918209734322, 3572106887472105263, 4692016570446185240464, 11496632576435936553085113, 52730955262459923752850296554, 454273406825238417871411598421653

Offset: 0

Andrew Howroyd, Dec 19 2020

nonn

The black nodes form an independent vertex set. For n > 0, a(n) is then the total number of indistinguishable independent vertex sets summed over distinct unlabeled graphs on n nodes.

Andrew Howroyd, Table of n, a(n) for n = 0..40
Eric Weisstein's World of Mathematics, Independent Vertex Set

A049312 counts bicolored graphs where adjacent nodes cannot have the same color.
A000666 counts bicolored graphs where adjacent nodes can have the same color.
Cf. A079491 (labeled case), A339830 (trees), A339836, A340021.

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}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
cross(u, v) = {sum(i=1, #u, sum(j=1, #v, gcd(u[i], v[j])))}
U(nb,nw)={my(s=0); forpart(v=nw, my(t=0); forpart(u=nb, t += permcount(u) * 2^cross(u,v)); s += t*permcount(v) * 2^edges(v)/nb!); s/nw!}
a(n) = {sum(k=0, n, U(k, n-k))}

A318870 Number of connected bipartite graphs on n unlabeled nodes with a distinguished bipartite block.

Original entry on oeis.org

1, 2, 1, 2, 4, 10, 27, 88, 328, 1460, 7799, 51196, 422521, 4483460, 62330116, 1150504224, 28434624153, 945480850638, 42417674401330, 2572198227615998, 211135833162079184, 23487811567341121158, 3545543330739039981738, 727053904070651775719646

Offset: 0

Andrew Howroyd, Sep 04 2018

nonn

Essentially the same as A007776. - Georg Fischer, Oct 02 2018

			a(1) = 2 because the single node can either be in the distinguished bipartite block or not.
a(2) = 1 because the only connected bipartite graph on two nodes is the complete graph on two nodes.
a(3) = 2 because the only connected bipartite graph on three nodes is the path graph on three nodes and there is a choice about which nodes are in the distinguished block.

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

Cf. A005142, A049312, A123549, A318869.

mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
b[d_] := Sum[A[n, d - n], {n, 0, d}];
Join[{1}, EULERi[Array[b, 23]]] (* Jean-François Alcover, Sep 13 2018, after Alois P. Heinz in A049312 *)

Inverse Euler transform of A049312.

A339836 Number of bicolored graphs on n unlabeled nodes such that every white node is adjacent to a black node.

Original entry on oeis.org

1, 1, 3, 10, 47, 296, 2970, 49604, 1484277, 81494452, 8273126920, 1552510549440, 538647737513260, 346163021846858368, 413301431190875322768, 920040760819708654610560, 3832780109273882704828352620, 29989833030101321999992097828464, 442280129125813382230656891568680400

Offset: 0

Andrew Howroyd, Dec 19 2020

nonn

The black nodes form a dominating set. For n > 0, a(n) is then the total number of indistinguishable dominating sets summed over distinct unlabeled graphs on n nodes.

Andrew Howroyd, Table of n, a(n) for n = 0..40
Eric Weisstein's World of Mathematics, Dominating Set

A049312 counts bicolored graphs where adjacent nodes cannot have the same color.
A000666 counts bicolored graphs where adjacent nodes can have the same color.
Cf. A339832, A339834 (trees), A340021.

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}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
dom(u, v) = {prod(i=1, #u, 2^sum(j=1, #v, gcd(u[i], v[j]))-1)}
U(nb,nw)={my(s=0); forpart(u=nw, my(t=0); forpart(v=nb, t += permcount(v) * 2^edges(v) * dom(u,v)); s += t*permcount(u) * 2^edges(u)/nb!); s/nw!}
a(n)={sum(k=0, n, U(k, n-k))}

A340021 Number of bicolored graphs on n unlabeled nodes such that black nodes are not adjacent to each other and every white node is adjacent to a black node.

Original entry on oeis.org

1, 1, 2, 5, 16, 66, 407, 3948, 66781, 2057140, 117820559, 12562407832, 2488441442819, 915216371901462, 625792587599236833, 797474948692631218674, 1899724021357155410243835, 8486672841492724213636009230, 71324140440429733888694354552551, 1131126439181050621704917376323373818

Offset: 0

Andrew Howroyd, Dec 30 2020

nonn

The black nodes form a maximal independent vertex set (or a set that is both independent and dominating). For n > 0, a(n) is then the total number of indistinguishable maximal independent vertex sets summed over distinct unlabeled graphs with n nodes.

Andrew Howroyd, Table of n, a(n) for n = 0..40
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set

A049312 counts bicolored graphs where adjacent nodes cannot have the same color.
A000666 counts bicolored graphs where adjacent nodes can have the same color.
Cf. A339832 (independent only), A339836 (dominating only), A339837 (trees).

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
dom[u_, v_] := Product[2^Sum[GCD[u[[i]], v[[j]]], {j, 1, Length[v]}] - 1, {i, 1, Length[u]}];
U[nb_, nw_] := Module[{s = 0}, Do[t = 0; Do[t += permcount[v]*dom[u, v], {v, IntegerPartitions[nb]}]; s += t*permcount[u]*2^edges[u]/nb!, {u, IntegerPartitions[nw]}]; s/nw!];
a[n_] := Sum[U[k, n - k], {k, 0, n}];
Array[a, 20] (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)

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}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
dom(u, v) = {prod(i=1, #u, 2^sum(j=1, #v, gcd(u[i], v[j]))-1)}
U(nb, nw)={my(s=0); forpart(u=nw, my(t=0); forpart(v=nb, t += permcount(v) * dom(u, v)); s += t*permcount(u) * 2^edges(u)/nb!); s/nw!}
a(n)={sum(k=0, n, U(k, n-k))}

cp's OEIS Frontend

A049311 Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations.

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A028657 Triangle read by rows: T(n,k) = number of n-node graphs with k nodes in distinguished bipartite block, k = 0..n.

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A048194 Total number of split graphs (chordal + chordal complement) on n vertices.

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A007776 Number of connected posets with n elements of height 1.

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A055192 Number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block, up to isomorphism.

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A056152 Triangular array giving number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block with k=1..n-1 vertices, up to isomorphism.

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A339832 Number of bicolored graphs on n unlabeled nodes such that black nodes are not adjacent to each other.

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A318870 Number of connected bipartite graphs on n unlabeled nodes with a distinguished bipartite block.

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