A028657 -id:A028657 - 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

A002724 Number of inequivalent n X n binary matrices, where equivalence means permutations of rows or columns.

Original entry on oeis.org

1, 2, 7, 36, 317, 5624, 251610, 33642660, 14685630688, 21467043671008, 105735224248507784, 1764356230257807614296, 100455994644460412263071692, 19674097197480928600253198363072, 13363679231028322645152300040033513414, 31735555932041230032311939400670284689732948

Offset: 0

N. J. A. Sloane

A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

Also, number of bipartite graphs with both partite sets of size n, one of which is marked. For connected bipartite graphs, see A363846. - Max Alekseyev, Jun 24 2023

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..26 from Alois P. Heinz)
Manuel Kauers and Jakob Moosbauer, Good pivots for small sparse matrices, arXiv:2006.01623 [cs.SC], 2020.
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]
Mathematics Stack Exchange, How many n-by-m binary matrices are there up to row and column permutations
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
Marko Riedel, Maple code with two different algorithms
M. Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005.
Index entries for sequences related to binary matrices
Index to number of inequivalent matrices modulo permutation of rows and columns

Cf. A002623, A002727, A006148, A002728, A002725, A052269, A052271, A052272, A091059, A363845, A363846, A000595.
Cf. A028657 (this sequence is the diagonal). - N. J. A. Sloane, Sep 01 2013
Column k=2 of A246106.

Maple
```
# See Marko Riedel link.
```

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[A[n, n], {n, 0, 15}] (* Jean-François Alcover, Aug 10 2018, after Alois P. Heinz *)

a(n) = A(n,n) \\ A defined in A028657. - Andrew Howroyd, Mar 01 2023

a(n) = Sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n} (fixA[s_1, s_2, ...;t_1, t_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...)) where fixA[...] = 2^Sum_{i, j>=1} (gcd(i, j)*s_i*t_j). - Christian G. Bower, Dec 18 2003

a(n) = A028657(2*n, n). - Max Alekseyev, Jun 24 2023

More terms from Vladeta Jovovic, Feb 04 2000

a(15) from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008

A246106 Number A(n,k) of inequivalent n X n matrices with entries from [k], where equivalence means permutations of rows or columns; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 7, 1, 0, 1, 4, 27, 36, 1, 0, 1, 5, 76, 738, 317, 1, 0, 1, 6, 175, 8240, 90492, 5624, 1, 0, 1, 7, 351, 57675, 7880456, 64796982, 251610, 1, 0, 1, 8, 637, 289716, 270656150, 79846389608, 302752867740, 33642660, 1, 0

Offset: 0

Alois P. Heinz, Aug 13 2014

			Square array A(n,k) begins:
  1, 1,    1,        1,           1,              1, ...
  0, 1,    2,        3,           4,              5, ...
  0, 1,    7,       27,          76,            175, ...
  0, 1,   36,      738,        8240,          57675, ...
  0, 1,  317,    90492,     7880456,      270656150, ...
  0, 1, 5624, 64796982, 79846389608, 20834113243925, ...

Columns k = 0-10 give: A000007, A000012, A002724, A052269, A052271, A052272, A246112, A246113, A246114, A246115, A246116.
Rows n = 0-10 give: A000012, A001477, A039623, A058001, A058002, A058003, A058004, A246108, A246109, A246110, A246111.
Main diagonal gives A246107.
A028657, A242106, A353585 are related tables.
Cf. A242095, A256069.

b:= proc(n, i) option remember; `if`(n=0, [[]],
      `if`(i<1, [], [b(n, i-1)[], seq(map(p->[p[], [i, j]],
       b(n-i*j, i-1))[], j=1..n/i)]))
    end:
A:= proc(n, k) option remember; add(add(k^add(add(i[2]*j[2]*
      igcd(i[1], j[1]), j=t), i=s) /mul(i[1]^i[2]*i[2]!, i=s)
      /mul(i[1]^i[2]*i[2]!, i=t), t=b(n$2)), s=b(n$2))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A246106(n,k)=A353585(k,n,n) \\ M. F. Hasler, May 01 2022

A(n,k) = Sum_{i=0..k} C(k,i) * A256069(n,i).

A(n,k) = Sum_{p,q in P(n)} k^Sum_{i in p, j in q} gcd(i, j) / (N(p)*N(q)) where N(p) = Product_{distinct parts x in p} x^m(x)*m(x)!, m(x) = multiplicity of x in p. - M. F. Hasler, Apr 30 2022 [corrected by Anders Kaseorg, Oct 04 2024]

A002727 Number of 3 X n binary matrices up to row and column permutations.

Original entry on oeis.org

1, 4, 13, 36, 87, 190, 386, 734, 1324, 2284, 3790, 6080, 9473, 14378, 21323, 30974, 44159, 61898, 85440, 116286, 156240, 207446, 272432, 354162, 456097, 582238, 737205, 926298, 1155567, 1431892, 1763074, 2157904, 2626276, 3179278, 3829294, 4590118, 5477081

Offset: 0

N. J. A. Sloane

Also, number of unlabeled bipartite graphs with three left vertices and n right vertices. - Yavuz Oruc, Jan 22 2018

			G.f. = 1 + 4*x + 13*x^2 + 36*x^3 + 87*x^4 + 190*x^5 + 386*x^6 + 734*x^7 + ...

A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
A. Atmaca, and A. Yavuz Oruç, On the size of two families of unlabeled bipartite graphs, AKCE International Journal of Graphs and Combinatorics, 2017.
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
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.
Index entries for sequences related to binary matrices
Index entries for linear recurrences with constant coefficients, signature (4, -4, -2, 2, 4, 3, -12, 3, 4, 2, -2, -4, 4, -1).
Index entries for two-way infinite sequences

Cf. A000601, A002623, A006148, A006381.

A row of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

I:=[1,4,13,36,87,190,386,734,1324,2284,3790,6080,9473, 14378]; [n le 14 select I[n] else 4*Self(n-1)-4*Self(n-2)-2*Self(n-3)+2*Self(n-4)+4*Self(n-5)+3*Self(n-6)-12*Self(n-7)+ 3*Self(n-8)+4*Self(n-9)+2*Self(n-10)-2*Self(n-11)-4*Self(n-12)+4*Self(n-13)-Self(n-14): n in [1..50]]; // Vincenzo Librandi, Oct 13 2015

CoefficientList[Series[(x^6+x^4+2x^3+x^2+1)/((1-x)^4(1-x^2)^2(1-x^3)^2),{x,0,40}],x] (* or *) LinearRecurrence[{4,-4,-2,2,4,3,-12,3,4,2,-2,-4,4,-1},{1,4,13,36,87,190,386,734,1324,2284,3790,6080,9473,14378},41] (* Harvey P. Dale, Nov 10 2011 *)
Table[Which[
Mod[n, 3] == 0,
1/6 (1/27 (54 + 45 n + 12 n^2 + n^3) + 1/320 (4 + n) *(225 + 15 (-1)^n + 352 n + 172 n^2 + 32 n^3 + 2 n^4) + Binomial[7 + n, 7]),
Mod[n, 3] == 1,
1/6 (1/27 (50 + 45 n + 12 n^2 + n^3) + 1/320 (4 + n) *(225 + 15 (-1)^n + 352 n + 172 n^2 + 32 n^3 + 2 n^4) + Binomial[7 + n, 7]),
Mod[n, 3] == 2,
1/6 (1/27 (28 + 39 n + 12 n^2 + n^3) + 1/320 (4 + n) *(225 + 15 (-1)^n + 352 n + 172 n^2 + 32 n^3 + 2 n^4) + Binomial[7 + n, 7])
], {n, 0, 100}] (* Yavuz Oruc, Jan 22 2018 *)

{a(n) = (6*n^7 + 168*n^6 + 2121*n^5 + 15540*n^4 + 70084*n^3 + 190512*n^2 + n*[284544, 281709, 277824, 281709, 284544, 274989][n%6+1]) \ 181440 + 1}; /* Michael Somos, Aug 22 2016 */

x='x+O('x^99); Vec((1+x^2+2*x^3+x^4+x^6)/((1-x)^2*((1-x)*(1-x^2)*(1-x^3))^2)) \\ Altug Alkan, Mar 03 2018

Vec(G(3, x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: (x^6+x^4+2*x^3+x^2+1)/((1-x)^4*(1-x^2)^2*(1-x^3)^2). - Vladeta Jovovic, Feb 04 2000.
a(0)=1, a(1)=4, a(2)=13, a(3)=36, a(4)=87, a(5)=190, a(6)=386, a(7)=734, a(8)=1324, a(9)=2284, a(10)=3790, a(11)=6080, a(12)=9473, a(13)=14378. For n>13, a(n)=4*a(n-1)-4*a(n-2)-2*a(n-3)+2*a(n-4)+4*a(n-5)+3*a(n-6)- 12*a(n-7)+ 3*a(n-8)+4*a(n-9)+2*a(n-10)-2*a(n-11)-4*a(n-12)+4*a(n-13)-a(n-14). - Harvey P. Dale, Nov 10 2011
a(n) = -a(-8 - n) for all n in Z. - Michael Somos, Aug 22 2016
From Yavuz Oruc, Jan 22 2018: (Start)
If n == 0 (mod 3) then a(n)=(1/6)*(binomial(n+7,7) + (3(n+4)(2n^4 + 32n^3 + 172n^2 + 352n + 15(-1)^n + 225))/960 + (2(n^3 + 12n^2 + 45n + 54))/54).
If n == 1 (mod 3) then a(n)=(1/6)*(binomial(n+7,7) + (3(n+4)(2n^4 + 32n^3 + 172n^2 + 352n + 15(-1)^n + 225))/960 + (2(n^3 + 12n^2 + 45n + 50))/54).
If n == 2 (mod 3) then a(n)=(1/6)*(binomial(n+7,7) + (3(n+4)(2n^4 + 32n^3 + 172n^2 + 352n + 15(-1)^n + 225))/960 + (2(n^3 + 12n^2 + 39n + 28))/54). (End)

More terms from Vladeta Jovovic, Feb 04 2000

Definition corrected by Max Alekseyev, Feb 05 2010

A005748 Number of n-covers of an unlabeled 7-set.

Original entry on oeis.org

1, 20, 348, 6093, 108182, 1890123, 31500927, 490890277, 7086257602, 94548676765, 1167995082810, 13406707973018, 143598707530374, 1441525802509250, 13619352767824724, 121574625625030584, 1029031775725235400, 8285521569322196569, 63648858792893665714

Offset: 1

N. J. A. Sloane, Simon Plouffe

nonn

Number of n X 7 binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Vladeta Jovovic, Binary matrices up to row and column permutations

A diagonal of A055080.

Vec(G(7, x) - G(6, x) + O(x^30)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

Corrected and extended by Vladeta Jovovic, Jun 13 2000

Terms a(17) and beyond from Andrew Howroyd, Feb 28 2023

A006148 Number of 4 X n binary matrices up to row and column permutations.

Original entry on oeis.org

1, 5, 22, 87, 317, 1053, 3250, 9343, 25207, 64167, 155004, 357009, 787586, 1670643, 3419552, 6774765, 13027340, 24372942, 44462456, 79240762, 138204782, 236258358, 396409924, 653639898, 1060379169, 1694174350, 2668300758, 4146300078, 6361709115, 9644583474

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
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.
Index entries for linear recurrences with constant coefficients, signature (6, -12, 6, 6, -6, 22, -54, 33, -4, 12, 60, -125, 54, -54, 70, 87, -132, 64, -132, 87, 70, -54, 54, -125, 60, 12, -4, 33, -54, 22, -6, 6, 6, -12, 6, -1).
Index entries for two-way infinite sequences

Cf. A002623, A002727, A006380.

A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

CoefficientList[Series[(x^20 - x^19 + 4 x^18 + 9 x^17 + 23 x^16 + 39 x^15 + 90 x^14 + 131 x^13 + 204 x^12 + 238 x^11 + 252 x^10 + 238 x^9 + 204 x^8 + 131 x^7 + 90 x^6 + 39 x^5 + 23 x^4 + 9 x^3 + 4 x^2 - x + 1)/((1 - x^4)^3 (1 - x^3)^4 (1 - x^2)^3 (1 - x)^6), {x, 0, 45}], x] (* Vincenzo Librandi, Oct 13 2015 *)
LinearRecurrence[{6,-12,6,6,-6,22,-54,33,-4,12,60,-125,54,-54,70,87,-132,64,-132,87,70,-54,54,-125,60,12,-4,33,-54,22,-6,6,6,-12,6,-1},{1,5,22,87,317,1053,3250,9343,25207,64167,155004,357009,787586,1670643,3419552,6774765,13027340,24372942,44462456,79240762,138204782,236258358,396409924,653639898,1060379169,1694174350,2668300758,4146300078,6361709115,9644583474,14456861538,21439125178,31471971903,45755970759,65915132560,94129925265},30] (* Harvey P. Dale, Jun 22 2021 *)

Vec(G(4, x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: (x^20 - x^19 + 4*x^18 + 9*x^17 + 23*x^16 + 39*x^15 + 90*x^14 + 131*x^13 + 204*x^12 + 238*x^11 + 252*x^10 + 238*x^9 + 204*x^8 + 131*x^7 + 90*x^6 + 39*x^5 + 23*x^4 + 9*x^3 + 4*x^2 - x + 1)/((1 - x^4)^3*(1 - x^3)^4*(1 - x^2)^3*(1 - x)^6). - Vladeta Jovovic, Feb 04 2000

More terms from Vladeta Jovovic, Feb 04 2000
Definition corrected by Max Alekseyev, Feb 05 2010
More terms from Vincenzo Librandi, Oct 13 2015

A055080 Triangle T(n,k) read by rows, giving number of k-member minimal covers of an unlabeled n-set, k=1..n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 9, 4, 1, 1, 9, 23, 17, 5, 1, 1, 12, 51, 65, 28, 6, 1, 1, 16, 103, 230, 156, 43, 7, 1, 1, 20, 196, 736, 863, 336, 62, 8, 1, 1, 25, 348, 2197, 4571, 2864, 664, 86, 9, 1, 1, 30, 590, 6093, 22952, 25326, 8609, 1229, 115, 10, 1, 1, 36, 960

Offset: 1

Vladeta Jovovic, Jun 13 2000

Also number of unlabeled split graphs on n vertices and with a k-element clique (cf. A048194).

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   3,   1;
  1,  6,   9,   4,   1;
  1,  9,  23,  17,   5,   1;
  1, 12,  51,  65,  28,   6,  1;
  1, 16, 103, 230, 156,  43,  7, 1;
  1, 20, 196, 736, 863, 336, 62, 8, 1;
  ...
There are four minimal covers of an unlabeled 3-set: one 1-cover {{1,2,3}}, two 2-covers {{1,2},{3}}, {{1,2},{1,3}} and one 3-cover {{1},{2},{3}}.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
Vladeta Jovovic, Binary matrices up to row and column permutations
G. F. Royle, Counting Set Covers and Split Graphs, J. Integer Seqs., 3 (2000), #00.2.6.
Eric Weisstein's World of Mathematics, Minimal covers

Row sums give A048194.
Columns are A000012, A087811, A005783, A005784, A005785 A005786, A055066.
Diagonals are A000012, A000027, A005744, A005745, A005746, A005771, A005747, A005748.
Cf. A035348 for labeled case.
Cf. A002620, A028657.

\\ Needs A(n,m) from A028657.
T(n,k) = A(n-k, k) - if(kAndrew Howroyd, Feb 28 2023

T(n,k) = A028657(n,k) - A028657(n-1,k). - Andrew Howroyd, Feb 28 2023

A049312 Number of graphs with a distinguished bipartite block, by number of vertices.

Original entry on oeis.org

1, 2, 4, 8, 17, 38, 94, 258, 815, 3038, 13804, 78760, 580456, 5647602, 73645352, 1297920850, 31031370360, 1007551636038, 44432872400460, 2661065508648436, 216457998880015366, 23920728651724212120, 3593384834863975164882, 734240676501745813835934

Offset: 0

Peter J. Cameron

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

Inverse Euler transform is A318870.

			a(2)=4: null graph with 0, 1 or 2 vertices in the distinguished block and complete graph with 1 vertex in distinguished block.

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

Alois P. Heinz, Table of n, a(n) for n = 0..40
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Karen L. Collins, Ann N. Trenk, Finding Balance: Split Graphs and Related Classes, arXiv:1706.03092 [math.CO], June 2017.
M. Guay-Paquet, A. H. Morales and E. Rowland, Structure and enumeration of (3+ 1)-free posets, arXiv preprint arXiv:1212.5356 [math.CO], 2012-2013. - From _N. J. A. Sloane_, Feb 01 2013
J. M. Troyka, Split graphs: combinatorial species and asymptotics, arXiv:1803.07248 [math.CO], 2018-2019.
J. M. Troyka, Split graphs: combinatorial species and asymptotics, Electron. J. Combin., 26 (2019), #P2.42.
E. M. Wright, The k-connectedness of bipartite graphs, J. Lond. Math. Soc. (2), 25 (1982), 7-12.

Row sums of A028657.

Cf. A048194, A318870.

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)):
a:= d-> add(A(n, d-n), n=0..d):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 01 2014

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}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)

a(n) ~ 1/n! A047863(n) = 1/n! Sum_{k=0..n} binomial(n,k) * 2^(k(n-k)) (see Wright; see also Thm. 3.7 of the Troyka link, which cites Wright). - Justin M. Troyka, Oct 29 2018

More terms from Vladeta Jovovic, Jun 17 2000

A242093 Number A(n,k) of inequivalent n X k binary matrices, where equivalence means permutations of rows or columns or the symbol set; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 5, 2, 1, 1, 3, 8, 8, 3, 1, 1, 3, 14, 18, 14, 3, 1, 1, 4, 20, 47, 47, 20, 4, 1, 1, 4, 30, 95, 173, 95, 30, 4, 1, 1, 5, 40, 200, 545, 545, 200, 40, 5, 1, 1, 5, 55, 367, 1682, 2812, 1682, 367, 55, 5, 1, 1, 6, 70, 674, 4745, 14386, 14386, 4745, 674, 70, 6, 1

Offset: 0

Alois P. Heinz, Aug 14 2014

			A(1,4) = 3: [0 0 0 0], [1 0 0 0], [1 1 0 0].
A(1,5) = 3: [0 0 0 0 0], [1 0 0 0 0], [1 1 0 0 0].
A(2,2) = 5:
  [0 0]  [1 0]  [1 1]  [1 0]  [1 0]
  [0 0], [0 0], [0 0], [1 0], [0 1].
A(3,2) = 8:
  [0 0]  [1 0]  [1 1]  [1 0]  [1 0]  [1 0]  [1 0]  [1 1]
  [0 0], [0 0], [0 0], [1 0], [0 1], [1 0], [0 1], [1 0].
  [0 0]  [0 0]  [0 0]  [0 0]  [0 0]  [1 0]  [1 0]  [0 0]
Square array A(n,k) begins:
  1, 1,  1,   1,    1,     1,       1,        1, ...
  1, 1,  2,   2,    3,     3,       4,        4, ...
  1, 2,  5,   8,   14,    20,      30,       40, ...
  1, 2,  8,  18,   47,    95,     200,      367, ...
  1, 3, 14,  47,  173,   545,    1682,     4745, ...
  1, 3, 20,  95,  545,  2812,   14386,    68379, ...
  1, 4, 30, 200, 1682, 14386,  126446,  1072086, ...
  1, 4, 40, 367, 4745, 68379, 1072086, 16821330, ...

Columns (or rows) k=0-10 give: A000012, A008619, A006918(n+1), A246148, A246149, A246150, A246151, A246152, A246153, A246154, A246155.
Main diagonal gives A091059.
Cf. A028657, A241956, A242095.

with(numtheory):
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(add(mul(mul(add(d*
      coeff(u, x, d), d=divisors(ilcm(i, j)))^(igcd(i, j)*
      coeff(s, x, i)*coeff(t, x, j)), j=1..degree(t)),
      i=1..degree(s))/mul(i^coeff(u, x, i)*coeff(u, x, i)!,
      i=1..degree(u))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,
      i=1..degree(t))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,
      i=1..degree(s)), u=b(2$2)), t=b(n$2)), s=b(k$2))
    end:
A:= (n, k)-> g(sort([n, k])[]):
seq(seq(A(n, d-n), n=0..d), d=0..12);

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[Sum[Product[Product[With[{gc = GCD[i, j]* Coefficient[s, x, i]*Coefficient[t, x, j]}, If[gc == 0, 1, Sum[d* Coefficient[u, x, d], {d, Divisors[LCM[i, j]]}]^gc]], {j, 1, Exponent[t, x]}],
{i, Exponent[s, x]}]/Product[i^Coefficient[u, x, i]*Coefficient[u, x, i]!,
{i, Exponent[u, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!,
{i, Exponent[t, x]}]/Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!,
{i, Exponent[s, x]}], {u, b[2, 2]}], {t, b[n, n]}], {s, b[k, k]}];
A[n_, k_] := g @@ Sort[{n, k}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Apr 25 2016, adapted from Maple, updated Jan 01 2021 *)

A005783 Number of 3-covers of an unlabeled n-set.

Original entry on oeis.org

1, 3, 9, 23, 51, 103, 196, 348, 590, 960, 1506, 2290, 3393, 4905, 6945, 9651, 13185, 17739, 23542, 30846, 39954, 51206, 64986, 81730, 101935, 126141, 154967, 189093, 229269, 276325, 331182, 394830, 468372, 553002, 650016, 760824, 886963

Offset: 0

N. J. A. Sloane

Equals first differences of A002727. - Vladeta Jovovic, May 24 2000

Number of 3 X n binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Stefano Spezia, Table of n, a(n) for n = 0..10000 (terms for n = 1..1000 from T. D. Noe)
R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
Masaaki Harada, Ken Saito, Binary linear complementary dual codes, arXiv:1802.06985 [math.CO], 2018.
Vladeta Jovovic, Binary matrices up to row and column permutations
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,-1,3,6,-6,-3,1,3,1,-3,1).

Column 3 of A055080.

Cf. A002727, A002620, A005784, A005785, A005786, A055066.

CoefficientList[Series[(x^6+x^4+2x^3+x^2+1)/((1-x^3)^2(1-x^2)^2 (1-x)^3),{x,0,50}],x] (* Harvey P. Dale, May 19 2011 *)

Vec(G(3, x)*(1 - x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: (x^6+x^4+2*x^3+x^2+1)/((1-x^3)^2*(1-x^2)^2*(1-x)^3).
a(n) ~ n^6/4320. - Stefano Spezia, Aug 08 2022
a(n) = n^6/4320 + 7*n^5/1440 + 79*n^4/1728 + 35*n^3/144 + 2939*n^2/4320 + 8863*n/8640 + 1 + (n/16 + 7/32)*floor(n/2) + (n/9 + 11/27)*floor(n/3) + floor((n+1)/3)/27. - Vaclav Kotesovec, Aug 09 2022

More terms from Vladeta Jovovic, May 24 2000

a(0) = 1 prepended by Stefano Spezia, Aug 09 2022

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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A002724 Number of inequivalent n X n binary matrices, where equivalence means permutations of rows or columns.

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A246106 Number A(n,k) of inequivalent n X n matrices with entries from [k], where equivalence means permutations of rows or columns; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A002727 Number of 3 X n binary matrices up to row and column permutations.

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A005748 Number of n-covers of an unlabeled 7-set.

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A006148 Number of 4 X n binary matrices up to row and column permutations.

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A055080 Triangle T(n,k) read by rows, giving number of k-member minimal covers of an unlabeled n-set, k=1..n.

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