%I #38 Feb 17 2024 08:17:42

%S 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,

%T 0,15,330,1581,3760,5715,6165,4945,2997,1365,455,105,15,1,0,0,0,0,315,

%U 4410,23604,73755,159390,259105,331716,343161,290745,202755,116175

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

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

%H Michael De Vlieger, <a href="/A054548/b054548.txt">Table of n, a(n) for n = 0..10700</a>

%H A. N. Bhavale and B. N. Waphare, <a href="https://ajc.maths.uq.edu.au/pdf/78/ajc_v78_p073.pdf">Basic retracts and counting of lattices</a>, Australasian J. of Combinatorics (2020) Vol. 78, No. 1, 73-99.

%H R. Tauraso, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Tauraso/tauraso18.html">Edge cover time for regular graphs</a>, JIS 11 (2008) 08.4.4.

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

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

%e From _Gus Wiseman_, Feb 14 2024: (Start)

%e Triangle begins:

%e 1

%e 0

%e 0 1

%e 0 0 3 1

%e 0 0 3 16 15 6 1

%e 0 0 0 30 135 222 205 120 45 10 1

%e Row n = 4 counts the following graphs:

%e . . 12-34 12-13-14 12-13-14-23 12-13-14-23-24 12-13-14-23-24-34

%e 13-24 12-13-24 12-13-14-24 12-13-14-23-34

%e 14-23 12-13-34 12-13-14-34 12-13-14-24-34

%e 12-14-23 12-13-23-24 12-13-23-24-34

%e 12-14-34 12-13-23-34 12-14-23-24-34

%e 12-23-24 12-13-24-34 13-14-23-24-34

%e 12-23-34 12-14-23-24

%e 12-24-34 12-14-23-34

%e 13-14-23 12-14-24-34

%e 13-14-24 12-23-24-34

%e 13-23-24 13-14-23-24

%e 13-23-34 13-14-23-34

%e 13-24-34 13-14-24-34

%e 14-23-24 13-23-24-34

%e 14-23-34 14-23-24-34

%e 14-24-34

%e (End)

%t 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 *)

%t 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 *)

%Y Row sums give A006129. Cf. A054547.

%Y The connected case is A062734, with loops A369195.

%Y This is the covering case of A084546.

%Y Column sums are A121251, with loops A173219.

%Y The version with loops is A369199, row sums A322661.

%Y The unlabeled version is A370167, row sums A002494.

%Y A006125 counts simple graphs; also loop-graphs if shifted left.

%Y Cf. A000666, A006649, A062740, A066383, A121252, A133686, A322700, A368597, A369196, A369197.

%K easy,nonn,tabf

%O 0,7

%A _Vladeta Jovovic_, Apr 09 2000

%E a(0) prepended by _Gus Wiseman_, Feb 14 2024