A000666 Number of symmetric relations on n nodes. (Formerly M1650 N0646)

1, 2, 6, 20, 90, 544, 5096, 79264, 2208612, 113743760, 10926227136, 1956363435360, 652335084592096, 405402273420996800, 470568642161119963904, 1023063423471189431054720, 4178849203082023236058229792, 32168008290073542372004082199424

Offset

0,2

Comments

Each node may or may not be related to itself.

Also the number of rooted graphs on n+1 nodes.

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.

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

References

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 101, 241.

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.

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

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

David Applegate, Table of n, a(n) for n = 0..80 [Shortened file because terms grow rapidly: see Applegate link below for additional terms]

David Applegate, Table of n, a(n) for n = 0..100

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

R. L. Davis, The number of structures of finite relations, Proc. Amer. Math. Soc. 4 (1953), 486-495.

R. L. Davis, Structure of dominance relations, Bull. Math. Biophys., 16 (1954), 131-140. [Annotated scanned copy]

F. Harary, The number of linear, directed, rooted, and connected graphs, Trans. Am. Math. Soc. 78 (1955) 445-463, eq. (24).

Frank Harary, Edgar M. Palmer, Robert W. Robinson and Allen J. Schwenk, Enumeration of graphs with signed points and lines, J. Graph Theory 1 (1977), no. 4, 295-308.

Adib Hassan, Po-Chien Chung and Wayne B. Hayes, Graphettes: Constant-time determination of graphlet and orbit identity including (possibly disconnected) graphlets up to size 8, arXiv:1708.04341 [cs.DS], 2017.

Mathematics Stack Exchange, Enumerating Graphs with Self-Loops, Jan 23 2014.

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, Sep. 15, 1955, pp. 14-22. [Annotated scanned copy]

W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.

W. Oberschelp, The number of non-isomorphic m-graphs, Presented at Mathematical Institute Oberwolfach, July 3 1967 [Scanned copy of manuscript]

W. Oberschelp, Strukturzahlen in endlichen Relationssystemen, in Contributions to Mathematical Logic (Proceedings 1966 Hanover Colloquium), pp. 199-213, North-Holland Publ., Amsterdam, 1968. [Annotated scanned copy]

G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

Marko Riedel, Maple code for sequence.

Marko Riedel et al., Enumerating Graphs with Self-Loops

R. W. Robinson, Notes - "A Present for Neil Sloane"

R. W. Robinson, Notes - computer printout

Sage, Common Graphs (Graph Generators)

Lorenzo Sauras-Altuzarra, Graphs

J. M. Tangen and N. J. A. Sloane, Correspondence, 1976-1976

Eric Weisstein's World of Mathematics, Rooted Graph

Formula

Euler transform of A054921. - N. J. A. Sloane, Oct 25 2018

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).

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]

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.

a(n) ~ 2^(n*(n+1)/2)/n! [McIlroy, 1955]. - Vaclav Kotesovec, Dec 19 2016

Maple

# see Riedel link above

Mathematica

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 *)
Table[Module[{eds, pms, leq},
eds=Select[Tuples[Range[n], 2], OrderedQ];
pms=Map[Sort, eds/.Table[i->Part[#, i], {i, n}]]&/@Permutations[Range[n]];
leq=Function[seq, PermutationCycles[Ordering[seq], Length]]/@pms;
Total[Thread[Power[2, leq]]]/n!
], {n, 0, 8}] (* This is after Geoffrey Critzer's program but does not use the (deprecated) Combinatorica package. - Gus Wiseman, Jul 21 2016 *)
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];
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]}];
a[n_] := a[n] = (s = 0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 17}] (* Jean-François Alcover, Nov 13 2017, after Andrew Howroyd *)

Prog

(PARI)
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}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2 + 1)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Oct 22 2017

Crossrefs

Cf. A000595, A001172, A001174, A006905, A000250, A054921 (connected relations).

Sequence in context: A124382 A318813 A336544 * A180890 A027321 A027315

Adjacent sequences: A000663 A000664 A000665 * A000667 A000668 A000669

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Description corrected by Christian G. Bower

More terms from Vladeta Jovovic, Apr 18 2000

Entry revised by N. J. A. Sloane, Mar 06 2007

Status

approved