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A002494
Number of n-node graphs without isolated nodes.
(Formerly M1762 N0699)
110
1, 0, 1, 2, 7, 23, 122, 888, 11302, 262322, 11730500, 1006992696, 164072174728, 50336940195360, 29003653625867536, 31397431814147073280, 63969589218557753586160, 245871863137828405125824848, 1787331789281458167615194471072, 24636021675399858912682459613241920
OFFSET
0,4
COMMENTS
Number of unlabeled simple graphs covering n vertices. - Gus Wiseman, Aug 02 2018
REFERENCES
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 214.
W. L. Kocay, Some new methods in reconstruction theory, Combinatorial Mathematics IX, 952 (1982) 89--114. [From Benoit Jubin, Sep 06 2008]
W. L. Kocay, On reconstructing spanning subgraphs, Ars Combinatoria, 11 (1981) 301--313. [From Benoit Jubin, Sep 06 2008]
J. H. Redfield, The theory of group-reduced distributions, Amer. J. Math., 49 (1927), 433-435; reprinted in P. A. MacMahon, Coll. Papers I, pp. 805-827.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
P. C. Fishburn and W. V. Gehrlein, Niche numbers, J. Graph Theory, 16 (1992), 131-139.
R. J. Mathar, Illustrations (2023)
J. H. Redfield, The theory of group-reduced distributions [Annotated scan of pages 452 and 453 only]
Eric Weisstein's World of Mathematics, Isolated Point.
Eric Weisstein's World of Mathematics, Graphical Partition
FORMULA
O.g.f.: (1-x)*G(x) where G(x) is o.g.f. for A000088. - Geoffrey Critzer, Apr 14 2012
a(n) = A327075(n)+A001349(n). - R. J. Mathar, Nov 21 2023
EXAMPLE
From Gus Wiseman, Aug 02 2018: (Start)
Non-isomorphic representatives of the a(4) = 7 graphs:
(12)(34)
(12)(13)(14)
(12)(13)(24)
(12)(13)(14)(23)
(12)(13)(24)(34)
(12)(13)(14)(23)(24)
(12)(13)(14)(23)(24)(34)
(End)
MAPLE
b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(ceil((p[j]-1)/2)
+add(igcd(p[k], p[j]), k=1..j-1), j=1..nops(p)))([l[], 1$n])),
add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
end:
a:= n-> b(n$2, [])-`if`(n>0, b(n-1$2, []), 0):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 14 2019
MATHEMATICA
<< MathWorld`Graphs`
Length /@ (gp = Select[ #, GraphicalPartitionQ] & /@
Graphs /@ Range[9])
nn = 20; g = Sum[NumberOfGraphs[n] x^n, {n, 0, nn}]; CoefficientList[Series[ g (1 - x), {x, 0, nn}], x] (*Geoffrey Critzer, Apr 14 2012*)
sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]], sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]], i}, {i, Length[Union@@m]}]], First[Sort[sysnorm[m, 1]]]];
sysnorm[m_, aft_]:=If[Length[Union@@m]<=aft, {m}, With[{mx=Table[Count[m, i, {2}], {i, Select[Union@@m, #>=aft&]}]}, Union@@(sysnorm[#, aft+1]&/@Union[Table[Map[Sort, m/.{par+aft-1->aft, aft->par+aft-1}, {0, 1}], {par, First/@Position[mx, Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Select[Subsets[Range[n]], Length[#]==2&]], Union@@#==Range[n]&]]], {n, 6}] (* Gus Wiseman, Aug 02 2018 *)
b[n_, i_, l_] := If[n==0 || i==1, 1/n!*2^(Function[p, Sum[Ceiling[(p[[j]]-1)/2] + Sum[GCD[p[[k]], p[[j]]], {k, 1, j-1}], {j, 1, Length[p]}]][Join[l, Table[1, {n}]]]), Sum[b[n-i*j, i-1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]];
a[n_] := b[n, n, {}] - If[n > 0, b[n-1, n-1, {}], 0];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 03 2019, after Alois P. Heinz *)
PROG
(Python)
from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A002494(n): return int(sum(Fraction(1<<sum(p[r]*p[s]*gcd(r, s) for r, s in combinations(p.keys(), 2))+sum((q>>1)*r+(q*r*(r-1)>>1) for q, r in p.items()), prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))-sum(Fraction(1<<sum(p[r]*p[s]*gcd(r, s) for r, s in combinations(p.keys(), 2))+sum((q>>1)*r+(q*r*(r-1)>>1) for q, r in p.items()), prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n-1))) if n else 1 # Chai Wah Wu, Jul 03 2024
CROSSREFS
Equals first differences of A000088. Cf. A006129 (labeled), A001349 (connected, inv. Euler Transf).
Sequence in context: A345871 A375130 A332802 * A032264 A139522 A163158
KEYWORD
nonn,nice
EXTENSIONS
More terms from Vladeta Jovovic, Apr 10 2000
a(0) added from David W. Wilson, Aug 24 2008
STATUS
approved