A188403 -id:A188403 - cp's OEIS frontend

A333737 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k up to permutations of rows and columns.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 5, 5, 1, 1, 1, 1, 3, 9, 12, 7, 1, 1, 1, 1, 4, 13, 33, 29, 11, 1, 1, 1, 1, 4, 20, 74, 142, 79, 15, 1, 1, 1, 1, 5, 28, 163, 556, 742, 225, 22, 1, 1, 1, 1, 5, 39, 319, 1919, 5369, 4454, 677, 30, 1, 1

Offset: 0

Andrew Howroyd, Apr 08 2020

Terms may be computed without generating each matrix by enumerating the number of matrices by column sum sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A188403. Burnside's lemma as applied in A318805 can be used to extend this method to the unlabeled case.

			Array begins:
==============================================
n\k | 0 1  2   3    4     5      6       7
----+-----------------------------------------
  0 | 1 1  1   1    1     1      1       1 ...
  1 | 1 1  1   1    1     1      1       1 ...
  2 | 1 1  2   2    3     3      4       4 ...
  3 | 1 1  3   5    9    13     20      28 ...
  4 | 1 1  5  12   33    74    163     319 ...
  5 | 1 1  7  29  142   556   1919    5793 ...
  6 | 1 1 11  79  742  5369  31781  156191 ...
  7 | 1 1 15 225 4454 64000 692599 5882230 ...
  ...
The T(3,3) = 5 matrices are:
   [0 0 3]  [0 1 2]  [0 1 2]  [1 0 2]  [1 1 1]
   [0 3 0]  [1 1 1]  [1 2 0]  [0 3 0]  [1 1 1]
   [3 0 0]  [2 1 0]  [2 0 1]  [2 0 1]  [1 1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..377
Ming Yean Lim, The number of irreducibles in the plethysm s_lambda[s_m], arXiv:2503.21108 [math.CO], 2025. See p. 8.

Rows n=0..5 are A000012, A000012, A008619, A106607, A333886, A333887.
Columns n=0..5 are A000012, A000012, A000041, A333888, A333889, A333890.
Main diagonal is A333738.
Cf. A188403 (labeled case), A333159 (binary), A333733 (not necessarily symmetric).
Cf. A318805, A333893.

A333893 Array read by antidiagonals: T(n,k) is the number of unlabeled loopless multigraphs with n nodes of degree k or less.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 5, 3, 1, 1, 1, 5, 8, 10, 3, 1, 1, 1, 6, 14, 26, 16, 4, 1, 1, 1, 7, 20, 61, 60, 29, 4, 1, 1, 1, 8, 30, 128, 243, 184, 45, 5, 1, 1, 1, 9, 40, 254, 800, 1228, 488, 75, 5, 1, 1, 1, 10, 55, 467, 2518, 7252, 6684, 1509, 115, 6, 1

Offset: 0

Andrew Howroyd, Apr 08 2020

T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k and isomorphism being up to simultaneous permutation of rows and columns. The case that allows independent permutations of rows and columns is covered by A333737.

Terms may be computed without generating each graph by enumerating the graphs by degree sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A188403. Burnside's lemma as applied in A192517 can be used to extend this method to the unlabeled case.

			Array begins:
==============================================
n\k | 0 1  2   3    4     5      6       7
----+-----------------------------------------
  0 | 1 1  1   1    1     1      1       1 ...
  1 | 1 1  1   1    1     1      1       1 ...
  2 | 1 2  3   4    5     6      7       8 ...
  3 | 1 2  5   8   14    20     30      40 ...
  4 | 1 3 10  26   61   128    254     467 ...
  5 | 1 3 16  60  243   800   2518    6999 ...
  6 | 1 4 29 184 1228  7252  38194  175369 ...
  7 | 1 4 45 488 6684 78063 772243 6254652 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..377 (antidiagonals 0..26)

Rows n=0..4 are A000012, A000012, A000027(n+1), A006918(n+1), A333897.
Columns k=0..5 are A000012, A008619, A000990, A333894, A333895, A333896.
Cf. A188403, A192517, A263293, A333161, A333330, A333737, A334546.

A188404 Number of (3*n) X n binary arrays with rows in nonincreasing order, 3 ones in every column and no more than 3 ones in any row.

Original entry on oeis.org

1, 4, 23, 214, 2698, 44288, 902962, 22262244, 648446612, 21940389584, 849992734124, 37273085398456, 1831837147680872, 100066601315825216, 6031974947471801512, 398733149802770699792, 28744536471179273843088, 2248840133521868856571456, 190105368229118222009348848

Offset: 1

R. H. Hardin, Mar 30 2011

nonn

Also, number of labeled graphs on n nodes with degree set {2,3}, with multiple edges and loops allowed. - N. J. A. Sloane, Sep 02 2013

			All solutions for 6 X 2:
..1..1....1..0....1..1....1..1
..1..1....1..0....1..0....1..1
..1..0....1..0....1..0....1..1
..0..1....0..1....0..1....0..0
..0..0....0..1....0..1....0..0
..0..0....0..1....0..0....0..0

Vaclav Kotesovec, Table of n, a(n) for n = 1..320
Frédéric Chyzak, Marni Mishna, Bruno Salvy, Effective Scalar Products for D-finite Symmetric Functions, arXiv:math/0310132v3 (version 2012)
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, pp. 584-585.
Goulden, I. P. and Jackson, D. M., Labelled graphs with small vertex degrees and $P$-recursiveness, SIAM J. Algebraic Discrete Methods 7(1986), no. 1, 60-66. MR0819706 (87k:05093).

Row 3 of A188403.

Cf. A005814, A228694.

max=20; f[x_]:=Sum[a[n]*(x^(n)/n!),{n,0,max}]; a[0]=1; a[1]=1; coef = CoefficientList[9*x^3*(x^4 - x^2 + x-2)*f''[x] - 3*(x^10 - 2*x^8 + 2*x^6 - 6*x^5 + 8*x^4 + 2*x^3 + 8*x^2 + 16*x - 8)*f'[x] + (x^11 + x^10 - 6*x^9 - 4*x^8 + 11*x^7 - 15*x^6 + 8*x^5 - 2*x^3 + 12*x^2 - 24*x - 24)*f[x],x]; Table[a[n],{n,0,max}]/.Solve[Thread[coef[[2;;max]]==0]][[1]]//Rest (* Vaclav Kotesovec, Sep 14 2014 *)
Flatten[{1,RecurrenceTable[{-(-7+n) * (-6+n) * (-5+n) * (-4+n) * (-3+n) * (-2+n) * (-1+n) * (-7+3 * n) * (4+114 * n-144 * n^2+27 * n^3) * a[-8+n]-(-6+n) * (-5+n) * (-4+n) * (-3+n) * (-2+n) * (-1+n) * (2+3 * n) * (-281+483 * n-225 * n^2+27 * n^3) * a[-7+n]+(-5+n) * (-4+n) * (-3+n) * (-2+n) * (-1+n) * (85-60 * n+9 * n^2) * (4+114 * n-144 * n^2+27 * n^3) * a[-6+n]+4 * (-4+n) * (-3+n) * (-2+n) * (-1+n) * (1112-3117 * n+2781 * n^2-864 * n^3+81 * n^4) * a[-5+n]-(-3+n) * (-2+n) * (-1+n) * (1820+4458 * n-14454 * n^2+10395 * n^3-2754 * n^4+243 * n^5) * a[-4+n]-3 * (-2+n) * (-1+n) * (-1892+6068 * n-7239 * n^2+3915 * n^3-945 * n^4+81 * n^5) * a[-3+n]-9 * (-1+n)^2 * (296+4904 * n-8256 * n^2+4563 * n^3-1026 * n^4+81 * n^5) * a[-2+n]-6 * (-728+9186 * n-16911 * n^2+10989 * n^3-2835 * n^4+243 * n^5) * a[-1+n]+12 * (-10+3 * n) * (-281+483 * n-225 * n^2+27 * n^3) * a[n]==0,a[2]==4,a[3]==23,a[4]==214,a[5]==2698,a[6]==44288,a[7]==902962,a[8]==22262244,a[9]==648446612},a,{n,2,20}]}] (* Vaclav Kotesovec, Sep 15 2014 *)

See Goulden and Jackson for the e.g.f. - N. J. A. Sloane, Sep 02 2013
Recurrence (for n>9): 12*(3*n - 10)*(27*n^3 - 225*n^2 + 483*n - 281)*a(n) = 6*(243*n^5 - 2835*n^4 + 10989*n^3 - 16911*n^2 + 9186*n - 728)*a(n-1) + 9*(n-1)^2*(81*n^5 - 1026*n^4 + 4563*n^3 - 8256*n^2 + 4904*n + 296)*a(n-2) + 3*(n-2)*(n-1)*(81*n^5 - 945*n^4 + 3915*n^3 - 7239*n^2 + 6068*n - 1892)*a(n-3) + (n-3)*(n-2)*(n-1)*(243*n^5 - 2754*n^4 + 10395*n^3 - 14454*n^2 + 4458*n + 1820)*a(n-4) - 4*(n-4)*(n-3)*(n-2)*(n-1)*(81*n^4 - 864*n^3 + 2781*n^2 - 3117*n + 1112)*a(n-5) - (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(9*n^2 - 60*n + 85)*(27*n^3 - 144*n^2 + 114*n + 4)*a(n-6) + (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(3*n + 2)*(27*n^3 - 225*n^2 + 483*n - 281)*a(n-7) + (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(3*n - 7)*(27*n^3 - 144*n^2 + 114*n + 4)*a(n-8). - Vaclav Kotesovec, Sep 14 2014
Asymptotics (Chyzak, 2003): a(n) ~ c * (n!)^(3/2) * (sqrt(3)/2)^n * exp(sqrt(3*n)) / n^(3/4), where c = 1/sqrt(2) * exp(3/4) / (2*Pi)^(3/4) = 0.37719937314536... . - Vaclav Kotesovec, Sep 14 2014

More terms from Vaclav Kotesovec, Sep 14 2014

A188400 Number of (6*n) X 6 binary arrays with rows in nonincreasing order, n ones in every column and no more than 2 ones in any row.

Original entry on oeis.org

1, 76, 2578, 44288, 478711, 3710272, 22393101, 111200600, 472211360, 1763603560, 5916521021, 18121655192, 51328536740, 135834620472, 338676787932, 801091475448, 1808025242415, 3912600581712, 8152010122276, 16411466716600, 32022153082291, 60720959942432, 112158519503545

Offset: 0

R. H. Hardin, Mar 30 2011

nonn

Number of 6 X 6 symmetric matrices with nonnegative integer entries and all row and column sums n. - Andrew Howroyd, Apr 07 2020

			Some solutions for 12X6
..1..0..0..1..0..0....1..0..0..0..1..0....1..0..0..1..0..0....1..1..0..0..0..0
..1..0..0..0..0..0....1..0..0..0..0..0....1..0..0..0..0..0....1..0..1..0..0..0
..0..1..0..0..0..0....0..1..1..0..0..0....0..1..1..0..0..0....0..1..1..0..0..0
..0..1..0..0..0..0....0..1..0..0..0..0....0..1..1..0..0..0....0..0..0..1..0..1
..0..0..1..1..0..0....0..0..1..1..0..0....0..0..0..1..0..0....0..0..0..1..0..1
..0..0..1..0..0..0....0..0..0..1..0..0....0..0..0..0..1..1....0..0..0..0..1..0
..0..0..0..0..1..1....0..0..0..0..1..0....0..0..0..0..1..0....0..0..0..0..1..0
..0..0..0..0..1..1....0..0..0..0..0..1....0..0..0..0..0..1....0..0..0..0..0..0
..0..0..0..0..0..0....0..0..0..0..0..1....0..0..0..0..0..0....0..0..0..0..0..0
..0..0..0..0..0..0....0..0..0..0..0..0....0..0..0..0..0..0....0..0..0..0..0..0
..0..0..0..0..0..0....0..0..0..0..0..0....0..0..0..0..0..0....0..0..0..0..0..0
..0..0..0..0..0..0....0..0..0..0..0..0....0..0..0..0..0..0....0..0..0..0..0..0

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

Column 6 of A188403.

a(0)=1 prepended and terms a(17) and beyond from Andrew Howroyd, Apr 07 2020

A188401 Number of (7*n) X 7 binary arrays with rows in nonincreasing order, n ones in every column and no more than 2 ones in any row.

Original entry on oeis.org

1, 232, 22054, 902962, 20758650, 313568636, 3444274966, 29445929253, 205617134345, 1214911142900, 6238854236951, 28428076580526, 116844463511314, 438968651731176, 1523790358321400, 4931487866349689, 14991891923697588, 43084786003434739, 117690415289036999

Offset: 0

R. H. Hardin, Mar 30 2011

nonn

Number of 7 X 7 symmetric matrices with nonnegative integer entries and all row and column sums n. - Andrew Howroyd, Apr 07 2020

			Some solutions for 14X7
..1..1..0..0..0..0..0....1..1..0..0..0..0..0....1..1..0..0..0..0..0
..1..0..0..0..0..0..0....1..0..0..0..0..0..1....1..0..0..1..0..0..0
..0..1..0..0..1..0..0....0..1..0..1..0..0..0....0..1..0..0..1..0..0
..0..0..1..1..0..0..0....0..0..1..0..0..0..0....0..0..1..0..0..0..0
..0..0..1..0..0..0..0....0..0..1..0..0..0..0....0..0..1..0..0..0..0
..0..0..0..1..0..0..0....0..0..0..1..0..1..0....0..0..0..1..0..1..0
..0..0..0..0..1..0..1....0..0..0..0..1..0..0....0..0..0..0..1..0..0
..0..0..0..0..0..1..0....0..0..0..0..1..0..0....0..0..0..0..0..1..0
..0..0..0..0..0..1..0....0..0..0..0..0..1..1....0..0..0..0..0..0..1
..0..0..0..0..0..0..1....0..0..0..0..0..0..0....0..0..0..0..0..0..1
..0..0..0..0..0..0..0....0..0..0..0..0..0..0....0..0..0..0..0..0..0
..0..0..0..0..0..0..0....0..0..0..0..0..0..0....0..0..0..0..0..0..0
..0..0..0..0..0..0..0....0..0..0..0..0..0..0....0..0..0..0..0..0..0
..0..0..0..0..0..0..0....0..0..0..0..0..0..0....0..0..0..0..0..0..0

Andrew Howroyd, Table of n, a(n) for n = 0..30

Column 7 of A188403.

a(0)=1 prepended and terms a(11) and beyond from Andrew Howroyd, Apr 07 2020

A188405 Number of (4*n) X n binary arrays with rows in nonincreasing order, 4 ones in every column and no more than 2 ones in any row.

Original entry on oeis.org

1, 1, 5, 42, 641, 14751, 478711, 20758650, 1158207312, 80758709676, 6877184737416, 701994697409136, 84574042067524470, 11870290445670605262, 1919446717950100963626, 354168049679464581788796, 73947210994621695613727526, 17342441149450781813176059990

Offset: 0

R. H. Hardin, Mar 30 2011

nonn

Number of n X n symmetric matrices with nonnegative integer entries and all row and column sums 4. - Andrew Howroyd, Apr 07 2020

In A005816 matrix elements on the diagonal are counted with a factor 2. This sequence here counts labeled multigraphs with n nodes (may be disconnected, undirected edges) without loops and degree at each node <=4. - R. J. Mathar, Jun 05 2022

			All solutions for 8X2
..1..1....1..1....1..1....1..0....1..1
..1..0....1..1....1..1....1..0....1..1
..1..0....1..1....1..0....1..0....1..1
..1..0....1..0....1..0....1..0....1..1
..0..1....0..1....0..1....0..1....0..0
..0..1....0..0....0..1....0..1....0..0
..0..1....0..0....0..0....0..1....0..0
..0..0....0..0....0..0....0..1....0..0

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

Row 4 of A188403.

Cf. A139670 (matrix elements 0 or 1).

a(0)=1 prepended and terms a(12) and beyond from Andrew Howroyd, Apr 07 2020

A188402 Number of (8*n) X 8 binary arrays with rows in nonincreasing order, n ones in every column and no more than 2 ones in any row.

Original entry on oeis.org

1, 764, 213798, 22262244, 1158207312, 36218801244, 767013376954, 11930327925108, 144413237202513, 1419823497519000, 11712930348839580, 83160597646878696, 518506187445244096, 2885792129983693112, 14530215365239964244, 66929085400566337832, 284683656715082259137

Offset: 0

R. H. Hardin, Mar 30 2011

nonn

Number of 8 X 8 symmetric matrices with nonnegative integer entries and all row and column sums n. - Andrew Howroyd, Apr 07 2020

			Some solutions for 16X8
..1..0..0..0..1..0..0..0....1..0..1..0..0..0..0..0....1..0..0..0..0..0..0..0
..1..0..0..0..1..0..0..0....1..0..0..1..0..0..0..0....1..0..0..0..0..0..0..0
..0..1..1..0..0..0..0..0....0..1..0..0..0..1..0..0....0..1..1..0..0..0..0..0
..0..1..0..0..0..0..1..0....0..1..0..0..0..0..0..0....0..1..0..0..0..0..0..0
..0..0..1..0..0..1..0..0....0..0..1..0..1..0..0..0....0..0..1..0..0..0..0..0
..0..0..0..1..0..0..1..0....0..0..0..1..0..0..0..0....0..0..0..1..0..0..0..1
..0..0..0..1..0..0..0..0....0..0..0..0..1..0..1..0....0..0..0..1..0..0..0..0
..0..0..0..0..0..1..0..0....0..0..0..0..0..1..0..1....0..0..0..0..1..0..1..0
..0..0..0..0..0..0..0..1....0..0..0..0..0..0..1..0....0..0..0..0..1..0..0..1
..0..0..0..0..0..0..0..1....0..0..0..0..0..0..0..1....0..0..0..0..0..1..1..0
..0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0....0..0..0..0..0..1..0..0
..0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0
..0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0
..0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0
..0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0
..0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0

Column 8 of A188403.

a(0)=1 prepended and terms a(9) and beyond from Andrew Howroyd, Apr 07 2020

A188406 Number of (5*n) X n binary arrays with rows in nonincreasing order, 5 ones in every column and no more than 2 ones in any row.

Original entry on oeis.org

1, 1, 6, 69, 1620, 62781, 3710272, 313568636, 36218801244, 5518184697792, 1078258692423672, 264082675302603684, 79537577428138854144, 28988656113108616134732, 12610397520409429024276176, 6469939930732195901177163384, 3874562148530193751600746715440, 2683567291732998871830525552934800

Offset: 0

R. H. Hardin, Mar 30 2011

nonn

Number of n X n symmetric matrices with nonnegative integer entries and all row and column sums 5. - Andrew Howroyd, Apr 07 2020

			All solutions for 10X2
..1..1....1..1....1..1....1..1....1..0....1..1
..1..1....1..0....1..1....1..1....1..0....1..1
..1..1....1..0....1..1....1..0....1..0....1..1
..1..1....1..0....1..1....1..0....1..0....1..0
..1..0....1..0....1..1....1..0....1..0....1..0
..0..1....0..1....0..0....0..1....0..1....0..1
..0..0....0..1....0..0....0..1....0..1....0..1
..0..0....0..1....0..0....0..1....0..1....0..0
..0..0....0..1....0..0....0..0....0..1....0..0
..0..0....0..0....0..0....0..0....0..1....0..0

Row 5 of A188403.

a(0)=1 prepended and terms a(10) and beyond from Andrew Howroyd, Apr 07 2020

A188407 Number of (6*n) X n binary arrays with rows in nonincreasing order, 6 ones in every column and no more than 2 ones in any row.

Original entry on oeis.org

1, 1, 7, 106, 3616, 222190, 22393101, 3444274966, 767013376954, 237357745294502, 98772845166962126, 53805754052806007528, 37518059561068661124848, 32857132982034738756201764, 35557273023840891076852950650, 46882317669656858737974124606500, 74388988137861405036759725929161668

Offset: 0

R. H. Hardin, Mar 30 2011

nonn

Number of n X n symmetric matrices with nonnegative integer entries and all row and column sums 6. - Andrew Howroyd, Apr 07 2020

			All solutions for 12X2
..1..0....1..1....1..1....1..1....1..1....1..1....1..1
..1..0....1..1....1..0....1..1....1..1....1..1....1..1
..1..0....1..1....1..0....1..0....1..1....1..1....1..1
..1..0....1..1....1..0....1..0....1..1....1..1....1..0
..1..0....1..0....1..0....1..0....1..1....1..1....1..0
..1..0....1..0....1..0....1..0....1..0....1..1....1..0
..0..1....0..1....0..1....0..1....0..1....0..0....0..1
..0..1....0..1....0..1....0..1....0..0....0..0....0..1
..0..1....0..0....0..1....0..1....0..0....0..0....0..1
..0..1....0..0....0..1....0..1....0..0....0..0....0..0
..0..1....0..0....0..1....0..0....0..0....0..0....0..0
..0..1....0..0....0..0....0..0....0..0....0..0....0..0

Row 6 of A188403.

a(0)=1 prepended and terms a(10) and beyond from Andrew Howroyd, Apr 06 2020

A188408 Number of (7*n) X n binary arrays with rows in nonincreasing order, 7 ones in every column and no more than 2 ones in any row.

Original entry on oeis.org

1, 1, 8, 154, 7340, 681460, 111200600, 29445929253, 11930327925108, 7061916086148812, 5884543061841730064, 6695191219474600587460, 10138670050730337096234992, 19996250425584243540741431296, 50416254105984511132422687541856, 159887463958073530310081258765881180

Offset: 0

R. H. Hardin, Mar 30 2011

nonn

Number of n X n symmetric matrices with nonnegative integer entries and all row and column sums 7. - Andrew Howroyd, Apr 07 2020

			All solutions for 14X2
..1..1....1..1....1..1....1..1....1..1....1..1....1..0....1..1
..1..1....1..1....1..1....1..1....1..1....1..0....1..0....1..1
..1..0....1..1....1..1....1..1....1..1....1..0....1..0....1..1
..1..0....1..1....1..1....1..1....1..1....1..0....1..0....1..0
..1..0....1..0....1..1....1..1....1..1....1..0....1..0....1..0
..1..0....1..0....1..1....1..0....1..1....1..0....1..0....1..0
..1..0....1..0....1..0....1..0....1..1....1..0....1..0....1..0
..0..1....0..1....0..1....0..1....0..0....0..1....0..1....0..1
..0..1....0..1....0..0....0..1....0..0....0..1....0..1....0..1
..0..1....0..1....0..0....0..0....0..0....0..1....0..1....0..1
..0..1....0..0....0..0....0..0....0..0....0..1....0..1....0..1
..0..1....0..0....0..0....0..0....0..0....0..1....0..1....0..0
..0..0....0..0....0..0....0..0....0..0....0..1....0..1....0..0
..0..0....0..0....0..0....0..0....0..0....0..0....0..1....0..0

Row 7 of A188403.

a(0)=1 prepended and terms a(9) and beyond from Andrew Howroyd, Apr 06 2020

cp's OEIS Frontend

A333737 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k up to permutations of rows and columns.

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A333893 Array read by antidiagonals: T(n,k) is the number of unlabeled loopless multigraphs with n nodes of degree k or less.

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A188404 Number of (3*n) X n binary arrays with rows in nonincreasing order, 3 ones in every column and no more than 3 ones in any row.

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A188400 Number of (6*n) X 6 binary arrays with rows in nonincreasing order, n ones in every column and no more than 2 ones in any row.

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A188401 Number of (7*n) X 7 binary arrays with rows in nonincreasing order, n ones in every column and no more than 2 ones in any row.

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A188405 Number of (4*n) X n binary arrays with rows in nonincreasing order, 4 ones in every column and no more than 2 ones in any row.

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A188402 Number of (8*n) X 8 binary arrays with rows in nonincreasing order, n ones in every column and no more than 2 ones in any row.

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A188406 Number of (5*n) X n binary arrays with rows in nonincreasing order, 5 ones in every column and no more than 2 ones in any row.

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A188407 Number of (6*n) X n binary arrays with rows in nonincreasing order, 6 ones in every column and no more than 2 ones in any row.

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