A333893 -id:A333893 - cp's OEIS frontend

A188403 T(n,k) = Number of (n*k) X k binary arrays with rows in nonincreasing order, n ones in every column and no more than 2 ones in any row.

1, 2, 1, 4, 3, 1, 10, 11, 4, 1, 26, 56, 23, 5, 1, 76, 348, 214, 42, 6, 1, 232, 2578, 2698, 641, 69, 7, 1, 764, 22054, 44288, 14751, 1620, 106, 8, 1, 2620, 213798, 902962, 478711, 62781, 3616, 154, 9, 1, 9496, 2313638, 22262244, 20758650, 3710272, 222190, 7340, 215, 10, 1

Offset: 1

R. H. Hardin, Mar 30 2011

From Andrew Howroyd, Apr 09 2020: (Start)
T(n,k) is the number of k X k symmetric matrices with nonnegative integer entries and all row and column sums n. The number of such matrices up to isomorphism is given in A333737.
T(n,k) is also the number of loopless multigraphs with k labeled nodes of degree n or less. The number of such multigraphs up to isomorphism is given in A333893. (End)

			Table starts
  1  2   4    10      26        76         232          764          2620
  1  3  11    56     348      2578       22054       213798       2313638
  1  4  23   214    2698     44288      902962     22262244     648446612
  1  5  42   641   14751    478711    20758650   1158207312   80758709676
  1  6  69  1620   62781   3710272   313568636  36218801244 5518184697792
  1  7 106  3616  222190  22393101  3444274966 767013376954 ...
  1  8 154  7340  681460 111200600 29445929253 ...
  1  9 215 13825 1865715 472211360 ...
  1 10 290 24510 4655535 ...
  1 11 381 41336 ...
  ...
All solutions for 4 X 2:
..1..0....1..1....1..1
..1..0....1..1....1..0
..0..1....0..0....0..1
..0..1....0..0....0..0

Andrew Howroyd, Table of n, a(n) for n = 1..351 (first 95 terms from R. H. Hardin; terms 96..153 from Alois P. Heinz)
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014

Columns 1..8 are A000012, A000027(n+1), A019298(n+1), A053493, A053494, A188400, A188401, A188402.
Rows 1..8 are A000085, A000985, A188404, A188405, A188406, A188407, A188408, A188409.
Main diagonal is A333739.
Cf. A257493, A333157, A333737, A333893.

T(k,n)={
  local(M=Map(Mat([0, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(r, h, p, q, v, e) = if(!p, acc(x^e+q, v), my(i=poldegree(p), t=pollead(p)); self()(r, k, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(j=1, min(t, (k-e)\m), self()(r, if(j==t, k, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e+j*m)))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n-r, k, src[i, 1], 0, src[i, 2], 0))); vecsum(Mat(M)[,2]);
}
{for(n=1, 7, for(k=1, 7, print1(T(n,k),", ")); print)} \\ Andrew Howroyd, Apr 08 2020

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.

Original entry on oeis.org

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.

A334546 Array read by antidiagonals: T(n,k) is the number of unlabeled connected loopless multigraphs with n nodes of degree k or less.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 0, 1, 1, 3, 2, 0, 0, 1, 1, 4, 4, 2, 0, 0, 1, 1, 5, 9, 12, 2, 0, 0, 1, 1, 6, 14, 37, 22, 2, 0, 0, 1, 1, 7, 23, 93, 146, 68, 2, 0, 0, 1, 1, 8, 32, 203, 602, 772, 166, 2, 0, 0, 1, 1, 9, 46, 399, 2126, 5847, 4449, 534, 2, 0, 0

Offset: 0

Andrew Howroyd, May 05 2020

This sequence may be derived from A333893 by inverse Euler transform.

			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 | 0 1 2   3    4     5      6       7 ...
  3 | 0 0 2   4    9    14     23      32 ...
  4 | 0 0 2  12   37    93    203     399 ...
  5 | 0 0 2  22  146   602   2126    6308 ...
  6 | 0 0 2  68  772  5847  34126  164965 ...
  7 | 0 0 2 166 4449 66289 716141 6021463 ...
  ...

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

Columns k=3..5 are A243391, A289157, A334547.
Main diagonal is A334546.
Cf. A289987, A328682 (regular), A333893 (not necessarily connected).

Column k is the inverse Euler transform of column k of A333893.

A333896 Number of unlabeled loopless multigraphs with n nodes of degree 5 or less.

Original entry on oeis.org

1, 1, 6, 20, 128, 800, 7252, 78063, 1059783, 17120947, 324989973, 7091405557, 175501697220, 4869895326986, 150149181850071, 5105014674244446, 190170557622677435, 7718536219718200652, 339658159537506154540, 16135241804080190069164, 824225772252119571338949

Offset: 0

Andrew Howroyd, Apr 08 2020

nonn

Column k=5 of A333893.

A333894 Number of unlabeled loopless multigraphs with n nodes of degree 3 or less.

Original entry on oeis.org

1, 1, 4, 8, 26, 60, 184, 488, 1509, 4468, 14494, 47228, 163584, 580489, 2155219, 8245929, 32708045, 133615420, 562518913, 2432472970, 10794454457, 49060744260, 228113422904, 1083584062308, 5253168117841, 25964917671574, 130732997627332, 669989546176410, 3492408687398352

Offset: 0

Andrew Howroyd, Apr 08 2020

nonn

Column k=3 of A333893.

Cf. A243391.

Euler transform of A243391.

A333895 Number of unlabeled loopless multigraphs with n nodes of degree 4 or less.

Original entry on oeis.org

1, 1, 5, 14, 61, 243, 1228, 6684, 42681, 304569, 2445296, 21676143, 210409238, 2213794888, 25064493886, 303468325469, 3909518220091, 53364222428953, 769009695515272, 11663042823897054, 185651491398956596, 3094082786005252494, 53871873427160960819

Offset: 0

Andrew Howroyd, Apr 08 2020

nonn

Column k=4 of A333893.

Cf. A289157.

Euler transform of A289157.

A333897 Number of unlabeled loopless multigraphs with 4 nodes of degree n or less.

Original entry on oeis.org

1, 3, 10, 26, 61, 128, 254, 467, 821, 1376, 2224, 3470, 5269, 7785, 11254, 15930, 22149, 30276, 40786, 54177, 71089, 92198, 118336, 150386, 189425, 236573, 293186, 360680, 440725, 535070, 645766, 774933, 925045, 1098666, 1298736, 1528308, 1790869, 2090007, 2429806, 2814476, 3248749

Offset: 0

Andrew Howroyd, Apr 08 2020

nonn

Row n=4 of A333893.

Conjectures from Colin Barker, Apr 10 2020: (Start)
G.f.: (1 + x + x^2 + x^3 + x^4)*(1 + x^2 + 2*x^3 + x^4 + x^6) / ((1 - x)^7*(1 + x)^3*(1 + x + x^2)^2).
a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3) - 4*a(n-4) + 2*a(n-5) + 9*a(n-6) - 9*a(n-8) - 2*a(n-9) + 4*a(n-10) + 4*a(n-11) - 2*a(n-12) - 2*a(n-13) + a(n-14) for n>13.
(End)

A333898 Number of unlabeled loopless multigraphs with n nodes of degree n or less.

Original entry on oeis.org

1, 1, 3, 8, 61, 800, 38194, 6254652, 3666019207, 7400038427299, 51429702889968135, 1241437835049586937641, 105176323921573775380216377, 31585284922954071317319026074538

Offset: 0

Andrew Howroyd, Apr 09 2020

Main diagonal of A333893.

cp's OEIS Frontend

A188403 T(n,k) = Number of (n*k) X k 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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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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A334546 Array read by antidiagonals: T(n,k) is the number of unlabeled connected loopless multigraphs with n nodes of degree k or less.

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A333896 Number of unlabeled loopless multigraphs with n nodes of degree 5 or less.

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A333894 Number of unlabeled loopless multigraphs with n nodes of degree 3 or less.

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A333895 Number of unlabeled loopless multigraphs with n nodes of degree 4 or less.

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A333897 Number of unlabeled loopless multigraphs with 4 nodes of degree n or less.

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A333898 Number of unlabeled loopless multigraphs with n nodes of degree n or less.

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