|
|
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)
|
|
16
|
|
|
1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, 816007449011040, 4374406209970747314, 64539836938720749739356, 2637796735571225009053373136, 300365896158980530053498490893399
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
FORMULA
|
There's an explicit formula - see for example Harary and Palmer (book), Eq. (5.4.14).
|
|
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!];
|
|
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
(Python)
from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A001174(n): return int(sum(Fraction(3**(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 15 2024
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,nice,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|