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

Author

Vladeta Jovovic, Apr 09 2000

Keywords

• easy
• nonn
• tabf

Examples

			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)
		

References

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

Links

• 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.

Crossrefs

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.

Programs

Mathematica

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 *)

Formula

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

Extensions

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