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

%I A054548 #42 Jan 12 2025 10:49:43
%S A054548 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 A054548 0,15,330,1581,3760,5715,6165,4945,2997,1365,455,105,15,1,0,0,0,0,315,
%U A054548 4410,23604,73755,159390,259105,331716,343161,290745,202755,116175
%N A054548 Triangular array giving number of labeled graphs on n unisolated nodes and k=0...n*(n-1)/2 edges.
%D A054548 F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, Page 29, Exercise 1.4.
%H A054548 Michael De Vlieger, <a href="/A054548/b054548.txt">Table of n, a(n) for n = 0..10700</a>
%H A054548 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 A054548 Ashok Nivrutti Bhavale, <a href="https://arxiv.org/abs/2501.05064">Equivalence of labeled graphs and lattices</a>, arXiv:2501.05064 [math.CO], 2025. See pp. 1-2, 13.
%H A054548 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 A054548 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 A054548 E.g.f.: exp(-x)*Sum_{n>=0} (1 + y)^C(n,2)*x^n/n!. - _Geoffrey Critzer_, Oct 07 2012
%e A054548 From _Gus Wiseman_, Feb 14 2024: (Start)
%e A054548 Triangle begins:
%e A054548    1
%e A054548    0
%e A054548    0   1
%e A054548    0   0   3   1
%e A054548    0   0   3  16  15   6   1
%e A054548    0   0   0  30 135 222 205 120  45  10   1
%e A054548 Row n = 4 counts the following graphs:
%e A054548   .  .  12-34  12-13-14  12-13-14-23  12-13-14-23-24  12-13-14-23-24-34
%e A054548         13-24  12-13-24  12-13-14-24  12-13-14-23-34
%e A054548         14-23  12-13-34  12-13-14-34  12-13-14-24-34
%e A054548                12-14-23  12-13-23-24  12-13-23-24-34
%e A054548                12-14-34  12-13-23-34  12-14-23-24-34
%e A054548                12-23-24  12-13-24-34  13-14-23-24-34
%e A054548                12-23-34  12-14-23-24
%e A054548                12-24-34  12-14-23-34
%e A054548                13-14-23  12-14-24-34
%e A054548                13-14-24  12-23-24-34
%e A054548                13-23-24  13-14-23-24
%e A054548                13-23-34  13-14-23-34
%e A054548                13-24-34  13-14-24-34
%e A054548                14-23-24  13-23-24-34
%e A054548                14-23-34  14-23-24-34
%e A054548                14-24-34
%e A054548 (End)
%t A054548 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 A054548 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 A054548 Row sums give A006129. Cf. A054547.
%Y A054548 The connected case is A062734, with loops A369195.
%Y A054548 This is the covering case of A084546.
%Y A054548 Column sums are A121251, with loops A173219.
%Y A054548 The version with loops is A369199, row sums A322661.
%Y A054548 The unlabeled version is A370167, row sums A002494.
%Y A054548 A006125 counts simple graphs; also loop-graphs if shifted left.
%Y A054548 Cf. A000666, A006649, A062740, A066383, A121252, A133686, A322700, A368597, A369196, A369197.
%K A054548 easy,nonn,tabf
%O A054548 0,7
%A A054548 _Vladeta Jovovic_, Apr 09 2000
%E A054548 a(0) prepended by _Gus Wiseman_, Feb 14 2024