%I A000666 M1650 N0646
%S A000666 1,2,6,20,90,544,5096,79264,2208612,113743760,10926227136,
%T A000666 1956363435360,652335084592096,405402273420996800,
%U A000666 470568642161119963904,1023063423471189431054720
%N A000666 Number of symmetric relations on n nodes.
%C A000666 Each node may or may not be related to itself.
%C A000666 Also the number of rooted graphs on n+1 nodes.
%C A000666 The 1-1-correspondence is as follows. Given a rooted graph on n+1 nodes, 
               replace each edge joining the root node to another node by a self-loop 
               at that node and erase the root node. The result is an undirected 
               graph on n nodes which is the graph of the symmetric relation.
%D A000666 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000666 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000666 R. L. Davies, The numbers of structures of finite relations, Proc. Amer. 
               Math. Soc., 4 (1953), 486-494.
%D A000666 F. Harary, The number of linear, directed, rooted and connected graphs, 
               Trans. Amer. Math. Soc., 78 (1955), 445-463.
%D A000666 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 
               1973, pp. 101, 241.
%D A000666 Harary, Frank; Palmer, Edgar M.; Robinson, Robert W.; Schwenk, Allen 
               J.; Enumeration of graphs with signed points and lines. J. Graph 
               Theory 1 (1977), no. 4, 295-308.
%D A000666 M. D. McIlroy, Calculation of numbers of structures of relations on finite 
               sets, Massachusetts Institute of Technology, Research Laboratory 
               of Electronics, Quarterly Progress Reports, No. 17, Sept. 15, 1955, 
               pp. 14-22.
%D A000666 W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. 
               Ann., 174 (1967), 53-78.
%D A000666 R. W. Robinson, Numerical implementation of graph counting algorithms, 
               AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%H A000666 David Applegate, <a href="b000666.txt">Table of n, a(n) for n = 0..80</
               a> [Shortened file because terms grow rapidly: see Applegate link 
               below for additional terms]
%H A000666 David Applegate, <a href="a000666.txt">Table of n, a(n) for n = 0..100</
               a>
%H A000666 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000666 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting 
               Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), 
               Article 04.3.2.
%H A000666 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RootedGraph.html">Rooted Graph</a>
%F A000666 Let G_{n+1,k} be the number of rooted graphs on n+1 nodes with k edges 
               and let G_{n+1}(x) = Sum_{k=0..n(n+1)/2} G_{n+1,k} x^k. Thus a(n) 
               = G_{n+1}(1). Let S_n(x_1, ..., x_n) denote the cycle index for Sym_n. 
               (cf. the link in A000142).
%F A000666 Compute x_1*S_n and regard it as the cycle index of a set of permutations 
               on n+1 points and find the corresponding cycle index for the action 
               on the n(n+1)/2 edges joining those points (the corresponding "pair 
               group"). Finally, by replacing each x_i by 1+x^i gives G_{n+1}(x). 
               [Harary]
%F A000666 Example, n=2. S_2 = (1/2)*(x_1^2+x_2), x_1*S_2 = (1/2)*(x_1^3+x_1*x_2). 
               The pair group is (1/2)*(x_1^2+x_1*x_2) and so G_3(x) = (1/2)*((1+x)^3+(1+x)*(1+x^2)) 
               = 1+2*x+2*x^2+x^3; set x=1 to get a(2) = 6.
%Y A000666 Cf. A000595, A001172, A001174, A006905, A000250.
%Y A000666 Sequence in context: A004104 A079468 A124382 this_sequence A027321 A027315 
               A005965
%Y A000666 Adjacent sequences: A000663 A000664 A000665 this_sequence A000667 A000668 
               A000669
%K A000666 nonn,nice
%O A000666 0,2
%A A000666 N. J. A. Sloane (njas(AT)research.att.com).
%E A000666 Description corrected by Christian Bower (bowerc(AT)usa.net). More terms 
               from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 18 2000
%E A000666 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Mar 06 2007