A100743 -id:A100743 - cp's OEIS frontend

A327377 Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly k endpoints (vertices of degree 1).

1, 0, 0, 0, 0, 1, 1, 0, 3, 0, 10, 12, 12, 4, 3, 253, 260, 160, 60, 35, 0, 12068, 9150, 4230, 1440, 480, 66, 15, 1052793, 570906, 195048, 53200, 12600, 2310, 427, 0, 169505868, 63523656, 15600032, 3197040, 585620, 95088, 14056, 1016, 105

Offset: 0

Gus Wiseman, Sep 05 2019

A graph is covering if there are no isolated vertices.

			Triangle begins:
      1
      0     0
      0     0     1
      1     0     3     0
     10    12    12     4     3
    253   260   160    60    35     0
  12068  9150  4230  1440   480    66    15

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Row sums are A006129.
Column k = 0 is A100743.
Column k = n is A123023.
Row sums without the first column are A327227.
The non-covering version is A327369.
The unlabeled version is A327372.
Cf. A004110, A006125, A055302, A055314, A059167, A245797, A327362, A327364, A327370.

Row(n)={ \\ R, U, B are e.g.f. of A055302, A055314, A059167.
  my(U=sum(n=2, n, x^n*sum(k=1, n, stirling(n-2, n-k, 2)*y^k/k!)) + O(x*x^n));
  my(R=sum(n=1, n, x^n*sum(k=1, n, stirling(n-1, n-k, 2)*y^k/k!)) + O(x*x^n));
  my(B=x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!)));
  my(A=exp(-x + O(x*x^n))*exp(x + U + subst(B-x, x, x*(1-y) + R)));
  Vecrev(n!*polcoef(A, n), n + 1);
}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Oct 05 2019

Column-wise inverse binomial transform of A327369.

E.g.f.: exp(-x)*exp(x + U(x,y) + B(x*(1-y) + R(x,y))), where R(x,y) is the e.g.f. of A055302, U(x,y) is the e.g.f. of A055314 and B(x) + x is the e.g.f. of A059167. - Andrew Howroyd, Oct 05 2019

Terms a(28) and beyond from Andrew Howroyd, Oct 05 2019

A324693 Number of simple graphs on n unlabeled nodes with minimum degree exactly 1.

Original entry on oeis.org

0, 1, 1, 4, 12, 60, 378, 3843, 64455, 1921532, 104098702, 10348794144, 1893781768084, 639954768875644, 400905675004630820, 467554784370658979194, 1019317687720204607541914, 4170177760438554428852944352, 32130458453030025927403299167172

Offset: 1

Andrew Howroyd, Sep 03 2019

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Minimum Vertex Degree
Gus Wiseman, The a(2) = 1 through a(5) = 12 unlabeled graphs with minimum degree 1.

Column k = 1 of A294217.
A diagonal of A263293.
The labeled version is A327227.
The generalization to set-systems is A327335, with covering case A327230.
Unlabeled covering graphs are A002494.
Cf. A000088, A004110, A100743, A141580, A245797, A261919, A327105, A327362, A327364, A327366, A327372.

a(n) = A002494(n) - A261919(n).

First differences of A141580. - Andrew Howroyd, Jan 11 2021

A369931 Triangle read by rows: T(n,k) is the number of labeled simple graphs with n edges and k vertices and without endpoints or isolated vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 6, 12, 0, 0, 0, 1, 85, 70, 0, 0, 0, 0, 100, 990, 465, 0, 0, 0, 0, 45, 2805, 11550, 3507, 0, 0, 0, 0, 10, 3595, 59990, 140420, 30016, 0, 0, 0, 0, 1, 2697, 147441, 1174670, 1802682, 286884, 0, 0, 0, 0, 0, 1335, 222516, 4710300, 22467312, 24556140, 3026655

Offset: 1

Andrew Howroyd, Feb 08 2024

T(n,k) is the number of traceless symmetric binary matrices with 2n 1's and k rows and at least two 1's in every row.

			Triangle begins:
  0;
  0, 0;
  0, 0, 1;
  0, 0, 0, 3;
  0, 0, 0, 6,  12;
  0, 0, 0, 1,  85,   70;
  0, 0, 0, 0, 100,  990,    465;
  0, 0, 0, 0,  45, 2805,  11550,    3507;
  0, 0, 0, 0,  10, 3595,  59990,  140420,   30016;
  0, 0, 0, 0,   1, 2697, 147441, 1174670, 1802682, 286884;
  ...
The T(3,3) = 1 matrix is:
  [0 1 1]
  [1 0 1]
  [1 1 0]
The T(4,4) = 3 matrices are:
  [0 0 1 1]  [0 1 0 1]  [0 1 1 0]
  [0 0 1 1]  [1 0 1 0]  [1 0 0 1]
  [1 1 0 0]  [0 1 0 1]  [1 0 0 1]
  [1 1 0 0]  [1 0 1 0]  [0 1 1 0]

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Row sums are A370059.
Column sums are A100743.
Main diagonal is A001205.
Cf. A369928, A369932 (unlabeled).

G(n)={my(A=x/exp(x*y + O(x*x^n))); exp(y*x^2/2 - x + O(x*x^n)) * sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2)*A^k/k!)}
T(n)={my(r=Vec(substvec(serlaplace(G(n)), [x, y], [y, x]))); vector(#r-1, i, Vecrev(Pol(r[i+1]/y), i))}

T(n,k) = k!*[x^k][y^n] exp(y*x^2/2 - x) * Sum_{j>=0} (1 + y)^binomial(j, 2)*(x/exp(y*x))^j/j!.

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A327377 Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly k endpoints (vertices of degree 1).

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A324693 Number of simple graphs on n unlabeled nodes with minimum degree exactly 1.

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A369931 Triangle read by rows: T(n,k) is the number of labeled simple graphs with n edges and k vertices and without endpoints or isolated vertices.

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