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A004103 Number of nets on n unlabeled nodes.
(Formerly M1942)
7

%I M1942 #31 Aug 17 2019 08:01:48

%S 1,2,9,56,705,19548,1419237,278474976,148192635483,213558945249402,

%T 836556995284293897,8962975658381123937708,

%U 264404516190234685662666051,21610417954162750247842392794292,4921335335427778307286708119839406529,3138313838161414849743136458064895837170596

%N Number of nets on n unlabeled nodes.

%C A net in this context is a graph with both signed vertices and signed edges. - _Andrew Howroyd_, Sep 25 2018

%D F. Harary and R. W. Robinson, Exposition of the enumeration of point-line-signed graphs, pp. 19 - 33 of Proc. Second Caribbean Conference Combinatorics and Computing (Bridgetown, 1977). Ed. R. C. Read and C. C. Cadogan. University of the West Indies, Cave Hill Campus, Barbados, 1977. vii+223 pp.

%D R. W. Robinson, personal communication.

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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Andrew Howroyd, <a href="/A004103/b004103.txt">Table of n, a(n) for n = 0..50</a> (terms 1..22 from R. W. Robinson)

%H Frank Harary, Edgar M. Palmer, Robert W. Robinson, 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 R. W. Robinson, <a href="/A000666/a000666_2.pdf">Notes - "A Present for Neil Sloane"</a>

%H R. W. Robinson, <a href="/A004102/a004102_1.pdf">Notes - computer printout</a>

%t 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 edges[v_] := Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[Quotient[v[[i]], 2], {i, 1, Length[v]}];

%t a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p]*2^Length[p], {p, IntegerPartitions[n]}]; s/n!];

%t Array[a, 16, 0] (* _Jean-François Alcover_, Aug 17 2019, after _Andrew Howroyd_ *)

%o (PARI)

%o 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 edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}

%o a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)*2^#p); s/n!} \\ _Andrew Howroyd_, Sep 25 2018

%Y Cf. A004102 (signed edges only), A000666 (signed vertices only).

%Y Cf. A004107.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E a(0)=1 prepended and a(13)-a(14) added by _Andrew Howroyd_, Sep 25 2018

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Last modified March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)