A001174 Number of oriented graphs (i.e., digraphs with no bidirected edges) on n unlabeled nodes. Also number of complete digraphs on n unlabeled nodes. Number of antisymmetric relations (i.e., oriented graphs with loops) on n unlabeled nodes is A083670. (Formerly M1809 N0715)

1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, 816007449011040, 4374406209970747314, 64539836938720749739356, 2637796735571225009053373136, 300365896158980530053498490893399

Offset

1,2

References

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 133, c_p.

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.

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

Andrew Howroyd, Table of n, a(n) for n = 1..50

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.

Musa Demirci, Ugur Ana, and Ismail Naci Cangul, Properties of Characteristic Polynomials of Oriented Graphs, Proc. Int'l Conf. Adv. Math. Comp. (ICAMC 2020) Springer, see p. 60.

F. Harary and E. M. Palmer, Enumeration of mixed graphs, Proc. Amer. Math. Soc., 17 (1966), 682-687.

T. R. Hoffman and J. P. Solazzo, Complex Two-Graphs via Equiangular Tight Frames, arXiv preprint arXiv:1408.0334 [math.CO], 2014-2017.

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]

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

Eric Weisstein's World of Mathematics, Oriented Graph

Formula

There's an explicit formula - see for example Harary and Palmer (book), Eq. (5.4.14).

a(n) ~ 3^(n*(n-1)/2)/n! [McIlroy, 1955]. - Vaclav Kotesovec, Dec 19 2016

Mathematica

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[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total @ Quotient[v - 1, 2];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!];
Array[a, 15] (* Jean-François Alcover, Jul 06 2018, 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]-1)\2)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ Andrew Howroyd, Oct 23 2017

Crossrefs

Cf. A000595, A001173, A281446.

Cf. A047656 (labeled case), A054941 (connected labeled case), A086345 (connected unlabeled case).

Sequence in context: A108042 A152559 A267239 * A067975 A065298 A091877

Adjacent sequences: A001171 A001172 A001173 * A001175 A001176 A001177

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic

Revised description from Vladeta Jovovic, Jan 20 2005

Status

approved