cp's OEIS Frontend

A000666 Number of symmetric relations on n nodes.

%I A000666 M1650 N0646 #136 Feb 16 2025 08:32:21
%S A000666 1,2,6,20,90,544,5096,79264,2208612,113743760,10926227136,
%T A000666 1956363435360,652335084592096,405402273420996800,
%U A000666 470568642161119963904,1023063423471189431054720,4178849203082023236058229792,32168008290073542372004082199424
%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-to-1 correspondence is as follows: Given a rooted graph on n+1 nodes, replace each edge joining the root node to another node with 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.
%C A000666 Also the number of the graphs with n nodes whereby each node is colored or not colored. A loop can be interpreted as a colored node. - _Juergen Will_, Oct 31 2011
%D A000666 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 101, 241.
%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 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%D A000666 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000666 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000666 David Applegate, <a href="/A000666/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/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/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000666 R. L. Davis, <a href="http://dx.doi.org/10.1090/S0002-9939-1953-0055294-2">The number of structures of finite relations</a>, Proc. Amer. Math. Soc. 4 (1953), 486-495.
%H A000666 R. L. Davis, <a href="/A000568/a000568_4.pdf">Structure of dominance relations</a>, Bull. Math. Biophys., 16 (1954), 131-140. [Annotated scanned copy]
%H A000666 F. Harary, <a href="http://dx.doi.org/10.1090/S0002-9947-1955-0068198-2">The number of linear, directed, rooted, and connected graphs</a>, Trans. Am. Math. Soc. 78 (1955) 445-463, eq. (24).
%H A000666 Frank Harary, Edgar M. Palmer, Robert W. Robinson and Allen J. Schwenk, <a href="http://dx.doi.org/10.1002/jgt.3190010405">Enumeration of graphs with signed points and lines</a>, J. Graph Theory 1 (1977), no. 4, 295-308.
%H A000666 Adib Hassan, Po-Chien Chung and Wayne B. Hayes, <a href="https://arxiv.org/abs/1708.04341">Graphettes: Constant-time determination of graphlet and orbit identity including (possibly disconnected) graphlets up to size 8</a>, arXiv:1708.04341 [cs.DS], 2017.
%H A000666 Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/41386/enumerating-graphs-with-self-loops">Enumerating Graphs with Self-Loops</a>, Jan 23 2014.
%H A000666 M. D. McIlroy, <a href="/A000088/a000088.pdf">Calculation of numbers of structures of relations on finite sets</a>, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sep. 15, 1955, pp. 14-22. [Annotated scanned copy]
%H A000666 W. Oberschelp, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002298732">Kombinatorische Anzahlbestimmungen in Relationen</a>, Math. Ann., 174 (1967), 53-78.
%H A000666 W. Oberschelp, <a href="/A000666/a000666_1.pdf">The number of non-isomorphic m-graphs</a>, Presented at Mathematical Institute Oberwolfach, July 3 1967 [Scanned copy of manuscript]
%H A000666 W. Oberschelp, <a href="/A000662/a000662.pdf"> Strukturzahlen in endlichen Relationssystemen</a>, in Contributions to Mathematical Logic (Proceedings 1966 Hanover Colloquium), pp. 199-213, North-Holland Publ., Amsterdam, 1968. [Annotated scanned copy]
%H A000666 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A000666 Marko Riedel, <a href="/A000666/a000666_2.maple.txt">Maple code for sequence.</a>
%H A000666 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/41386/">Enumerating Graphs with Self-Loops</a>
%H A000666 R. W. Robinson, <a href="/A000666/a000666_2.pdf">Notes - "A Present for Neil Sloane"</a>
%H A000666 R. W. Robinson, <a href="/A004102/a004102_1.pdf">Notes - computer printout</a>
%H A000666 Sage, <a href="http://www.sagemath.org/doc/reference/graphs/sage/graphs/graph_generators.html">Common Graphs (Graph Generators)</a>
%H A000666 Lorenzo Sauras-Altuzarra, <a href="/A000666/a000666.png">Graphs</a>
%H A000666 J. M. Tangen and N. J. A. Sloane, <a href="/A000666/a000666.pdf">Correspondence, 1976-1976</a>
%H A000666 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RootedGraph.html">Rooted Graph</a>
%F A000666 Euler transform of A054921. - _N. J. A. Sloane_, Oct 25 2018
%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.
%F A000666 a(n) ~ 2^(n*(n+1)/2)/n! [McIlroy, 1955]. - _Vaclav Kotesovec_, Dec 19 2016
%p A000666 # see Riedel link above
%t A000666 Join[{1,2}, Table[CycleIndex[Join[PairGroup[SymmetricGroup[n]], Permutations[Range[n*(n-1)/2+1, n*(n+1)/2]], 2],s] /. Table[s[i]->2, {i,1,n^2-n}], {n,2,8}]] (* _Geoffrey Critzer_, Nov 04 2011 *)
%t A000666 Table[Module[{eds,pms,leq},
%t A000666 eds=Select[Tuples[Range[n],2],OrderedQ];
%t A000666 pms=Map[Sort,eds/.Table[i->Part[#,i],{i,n}]]&/@Permutations[Range[n]];
%t A000666 leq=Function[seq,PermutationCycles[Ordering[seq],Length]]/@pms;
%t A000666 Total[Thread[Power[2,leq]]]/n!
%t A000666 ],{n,0,8}] (* This is after _Geoffrey Critzer_'s program but does not use the (deprecated) Combinatorica package. - _Gus Wiseman_, Jul 21 2016 *)
%t A000666 permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
%t A000666 edges[v_] := Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i-1}], {i, 2, Length[v]}] + Sum[Quotient[v[[i]], 2] + 1, {i, 1, Length[v]}];
%t A000666 a[n_] := a[n] = (s = 0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
%t A000666 Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 17}] (* _Jean-François Alcover_, Nov 13 2017, after _Andrew Howroyd_ *)
%o A000666 (PARI)
%o A000666 permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
%o A000666 edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, v[i]\2 + 1)}
%o A000666 a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ _Andrew Howroyd_, Oct 22 2017
%o A000666 (Python)
%o A000666 from itertools import combinations
%o A000666 from math import prod, factorial, gcd
%o A000666 from fractions import Fraction
%o A000666 from sympy.utilities.iterables import partitions
%o A000666 def A000666(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)+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))) # _Chai Wah Wu_, Jul 02 2024
%Y A000666 Cf. A000595, A001172, A001174, A006905, A000250, A054921 (connected relations).
%K A000666 nonn,nice
%O A000666 0,2
%A A000666 _N. J. A. Sloane_
%E A000666 Description corrected by _Christian G. Bower_
%E A000666 More terms from _Vladeta Jovovic_, Apr 18 2000
%E A000666 Entry revised by _N. J. A. Sloane_, Mar 06 2007