A054547 -id:A054547 - cp's OEIS frontend

A054548 Triangular array giving number of labeled graphs on n unisolated nodes and k=0...n*(n-1)/2 edges.

1, 0, 0, 1, 0, 0, 3, 1, 0, 0, 3, 16, 15, 6, 1, 0, 0, 0, 30, 135, 222, 205, 120, 45, 10, 1, 0, 0, 0, 15, 330, 1581, 3760, 5715, 6165, 4945, 2997, 1365, 455, 105, 15, 1, 0, 0, 0, 0, 315, 4410, 23604, 73755, 159390, 259105, 331716, 343161, 290745, 202755, 116175

Offset: 0

Vladeta Jovovic, Apr 09 2000

			From _Gus Wiseman_, Feb 14 2024: (Start)
Triangle begins:
   1
   0
   0   1
   0   0   3   1
   0   0   3  16  15   6   1
   0   0   0  30 135 222 205 120  45  10   1
Row n = 4 counts the following graphs:
  .  .  12-34  12-13-14  12-13-14-23  12-13-14-23-24  12-13-14-23-24-34
        13-24  12-13-24  12-13-14-24  12-13-14-23-34
        14-23  12-13-34  12-13-14-34  12-13-14-24-34
               12-14-23  12-13-23-24  12-13-23-24-34
               12-14-34  12-13-23-34  12-14-23-24-34
               12-23-24  12-13-24-34  13-14-23-24-34
               12-23-34  12-14-23-24
               12-24-34  12-14-23-34
               13-14-23  12-14-24-34
               13-14-24  12-23-24-34
               13-23-24  13-14-23-24
               13-23-34  13-14-23-34
               13-24-34  13-14-24-34
               14-23-24  13-23-24-34
               14-23-34  14-23-24-34
               14-24-34
(End)

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, Page 29, Exercise 1.4.

Michael De Vlieger, Table of n, a(n) for n = 0..10700
A. N. Bhavale and B. N. Waphare, Basic retracts and counting of lattices, Australasian J. of Combinatorics (2020) Vol. 78, No. 1, 73-99.
Ashok Nivrutti Bhavale, Equivalence of labeled graphs and lattices, arXiv:2501.05064 [math.CO], 2025. See pp. 1-2, 13.
R. Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4.

Row sums give A006129. Cf. A054547.
The connected case is A062734, with loops A369195.
This is the covering case of A084546.
Column sums are A121251, with loops A173219.
The version with loops is A369199, row sums A322661.
The unlabeled version is A370167, row sums A002494.
A006125 counts simple graphs; also loop-graphs if shifted left.
Cf. A000666, A006649, A062740, A066383, A121252, A133686, A322700, A368597, A369196, A369197.

nn=5; s=Sum[(1+y)^Binomial[n,2]  x^n/n!, {n,0,nn}]; Range[0,nn]! CoefficientList[Series[ s Exp[-x], {x,0,nn}], {x,y}] //Grid  (* returns triangle indexed at n = 0, Geoffrey Critzer, Oct 07 2012 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}],{k}],Union@@#==Range[n]&]],{n,0,5},{k,0,Binomial[n,2]}] (* Gus Wiseman, Feb 14 2024 *)

T(n, k) = Sum_{i=0..n} (-1)^(n-i)*C(n, i)*C(C(i, 2), k), k=0...n*(n-1)/2.

E.g.f.: exp(-x)*Sum_{n>=0} (1 + y)^C(n,2)*x^n/n!. - Geoffrey Critzer, Oct 07 2012

a(0) prepended by Gus Wiseman, Feb 14 2024

A370167 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with k = 0..binomial(n,2) edges.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 1, 1, 0, 0, 0, 1, 4, 5, 5, 4, 2, 1, 1, 0, 0, 0, 1, 3, 9, 15, 20, 22, 20, 14, 9, 5, 2, 1, 1, 0, 0, 0, 0, 1, 6, 20, 41, 73, 110, 133, 139, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1, 0, 0, 0, 0, 1, 3, 15, 50, 124, 271, 515, 832, 1181, 1460, 1581, 1516, 1291, 970, 658, 400, 220, 114, 56, 24, 11, 5, 2, 1, 1

Offset: 0

Gus Wiseman, Feb 15 2024

			Triangle begins:
  1
  0
  0  1
  0  0  1  1
  0  0  1  2  2  1  1
  0  0  0  1  4  5  5  4  2  1  1
  0  0  0  1  3  9 15 20 22 20 14  9  5  2  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..1350 (rows 0..20)

Column sums are A000664.
Row sums are A002494.
This is the covering case of A008406, labeled A084546.
The labeled version is A054548, row sums A006129, column sums A121251.
The connected case is A054924, row sums A001349, column sums A002905.
The labeled connected case is A062734, with loops A369195.
The connected case with loops is A283755, row sums A054921.
The labeled version w/ loops is A369199, row sums A322661, col sums A173219.
Cf. A000666, A006125, A006649, A054547, A066383, A322700.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{2}],{k}],Union@@#==Range[n]&]]], {n,0,5},{k,0,Binomial[n,2]}]

\\ G(n) defined in A008406.
row(n)={Vecrev(G(n)-if(n>0, G(n-1)), binomial(n,2)+1)}
{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Feb 19 2024

a(42) onwards from Andrew Howroyd, Feb 19 2024

A121252 Number of labeled digraphs without isolated vertices and with n arcs.

Original entry on oeis.org

1, 2, 25, 520, 15150, 567540, 25986334, 1406214816, 87803248260, 6213408758960, 491423381371208, 42958461746823376, 4112928296979824348, 428022362222941505760, 48106759414197240877200, 5807382441785322296139392

Offset: 0

Vladeta Jovovic, Aug 22 2006

Cf. A054545, A053418 (unlabeled).

seq(value(Sum(binomial(m*(m-1),n)/2^(m+1),m=0..infinity)),n=0..20); # Vladeta Jovovic, Aug 25 2006

a(n) = Sum_{m>=0} binomial(m*(m-1),n)/2^(m+1). Column sums of A054547.

More terms from Max Alekseyev, Aug 23 2006

A350908 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled unisolated nodes with k arcs, k = 0..n*(n-1).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 3, 4, 4, 1, 1, 0, 0, 1, 9, 23, 37, 47, 38, 27, 13, 5, 1, 1, 0, 0, 0, 3, 34, 116, 331, 669, 1128, 1477, 1665, 1489, 1154, 707, 379, 154, 61, 16, 5, 1, 1, 0, 0, 0, 1, 15, 134, 664, 2535, 7796, 19719, 42193, 77324, 122960, 170317, 206983, 220768

Offset: 1

Andrew Howroyd, Jan 28 2022

			Triangle begins:
  [1] 0;
  [2] 0, 1, 1;
  [3] 0, 0, 3, 4,  4,  1,  1;
  [4] 0, 0, 1, 9, 23, 37, 47, 38, 27, 13, 5, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)

Row sums are A053598.
Column sums are A053418.
The labeled version is A054547.
Cf. A052283, A054733.

\\ See A054733 for G.
row(n)={Vecrev(G(n,y)-G(n-1,y), n*(n-1)+1)}
{ for(n=1, 6, print(row(n))) }

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A054548 Triangular array giving number of labeled graphs on n unisolated nodes and k=0...n*(n-1)/2 edges.

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A370167 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with k = 0..binomial(n,2) edges.

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A121252 Number of labeled digraphs without isolated vertices and with n arcs.

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A350908 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled unisolated nodes with k arcs, k = 0..n*(n-1).

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