cp's OEIS Frontend

A052283 Triangle read by rows: T(n,k) is the number of unlabeled directed graphs on n nodes with k arcs, k=0..n*(n-1).

%I A052283 #45 Feb 16 2025 08:32:41
%S A052283 1,1,1,1,1,1,1,4,4,4,1,1,1,1,5,13,27,38,48,38,27,13,5,1,1,1,1,5,16,61,
%T A052283 154,379,707,1155,1490,1670,1490,1155,707,379,154,61,16,5,1,1,1,1,5,
%U A052283 17,76,288,1043,3242,8951,21209,43863,78814,124115,171024,207362,220922,207362,171024,124115,78814,43863,21209,8951,3242,1043,288,76,17,5,1,1
%N A052283 Triangle read by rows: T(n,k) is the number of unlabeled directed graphs on n nodes with k arcs, k=0..n*(n-1).
%C A052283 Triangular array read by rows T(n,k) is the number of unlabeled directed graphs (no self loops allowed) on n nodes with exactly k edges where n >= 1, 0 <= k <= n(n-1). - _Geoffrey Critzer_, Nov 01 2011
%D A052283 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 247.
%D A052283 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 522.
%H A052283 Andrew Howroyd, <a href="/A052283/b052283.txt">Table of n, a(n) for n = 0..2680</a> (rows 0..20)
%H A052283 R. Absil and H Mélot, <a href="http://arxiv.org/abs/1304.7993">Digenes: genetic algorithms to discover conjectures about directed and undirected graphs</a>, arXiv preprint arXiv:1304.7993 [cs.DM], 2013.
%H A052283 Philippe Ramirez and Stéphane Legendre, <a href="https://doi.org/10.1016/j.socnet.2017.09.004">Revisiting asymmetric marriage rules</a>, in Social Networks 52 (2017), pp. 261-269.
%H A052283 Stackexchange, <a href="http://math.stackexchange.com/questions/2187019/number-of-distinct-connected-digraphs-with-4-vertices-and-6-edges">Number of distinct connected digraphs...</a>, (2017)
%H A052283 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SimpleDirectedGraph.html">Simple Directed Graph</a>
%F A052283 T(n,0) = T(n,1) = T(n,n(n-1)-1) = T(n,n) = 1. - _Geoffrey Critzer_, Nov 01 2011
%F A052283 T(2k,k) = T(2k+1,k) = T(2k+2,k) =... and is the maximum value of column k. - _Geoffrey Critzer_, Nov 01 2011
%e A052283 [1],
%e A052283 [1],
%e A052283 [1,1,1],
%e A052283 [1,1,4,4,4,1,1],
%e A052283 [1,1,5,13,27,38,48,38,27,13,5,1,1];
%e A052283 (the last batch giving the numbers of directed graphs with 4 nodes and from 0 to 12 arcs).
%t A052283 Table[CoefficientList[GraphPolynomial[n, x, Directed], x], {n, 1, 10}] (* _Geoffrey Critzer_, Nov 01 2011 *)
%t A052283 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 A052283 edges[v_, t_] := Product[g = GCD[v[[i]], v[[j]]]; t[v[[i]]*v[[j]]/g]^(2 g), {i, 2, Length[v]}, {j, 1, i-1}] * Product[ t[v[[i]]]^(v[[i]] - 1), {i, 1, Length[v]}];
%t A052283 gp[n_] := (s = 0; Do[s += permcount[p]*edges[p, 1 + x^# &], {p, IntegerPartitions[n]}]; s/n!);
%t A052283 A052283 = Reap[For[n = 1, n <= 6, n++, p = gp[n]; For[k = 0, k <= Exponent[p, x], k++, Sow[Coefficient[p, x, k]]]]][[2, 1]] (* _Jean-François Alcover_, Jul 09 2018, after _Andrew Howroyd_ *)
%o A052283 (PARI)
%o A052283 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 A052283 edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^(2*g))) * prod(i=1, #v, t(v[i])^(v[i]-1))}
%o A052283 gp(n) = {my(s=0); forpart(p=n, s+=permcount(p)*edges(p,i->1+x^i)); s/n!}
%o A052283 for(n=1, 6, my(p=gp(n)); for(k=0, poldegree(p), print1(polcoeff(p,k), ", ")); print); \\ _Andrew Howroyd_, Nov 05 2017
%Y A052283 Cf. A000273 (row sums), A070166, A008406, A003085, A283753 (weakly connected).
%K A052283 nonn,tabf
%O A052283 0,8
%A A052283 _Vladeta Jovovic_, Feb 07 2000
%E A052283 a(0)=1 prepended and terms a(62) and beyond from _Andrew Howroyd_, Apr 20 2020