

A004106


Number of lineselfdual nets (or edgeselfdual nets) with n nodes.
(Formerly M0889)


3



1, 2, 3, 8, 29, 148, 1043, 11984, 229027, 6997682, 366204347, 30394774084, 4363985982959, 994090870519508, 393850452332173999, 249278602955869472540, 275042591834324901085904, 488860279973733024992540668, 1514493725905920009795681408275
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

A net in this context is a graph with both signed vertices and signed edges. A net is lineselfdual if changing the signs on all edges leaves the graph unchanged up to isomorphism.  Andrew Howroyd, Sep 25 2018


REFERENCES

F. Harary and R. W. Robinson, Exposition of the enumeration of pointlinesigned 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.
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
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 = 0..50 (terms 1..22 from R. W. Robinson)
Frank Harary, Edgar M. Palmer, Robert W. Robinson, Allen J. Schwenk, Enumeration of graphs with signed points and lines, J. Graph Theory 1 (1977), no. 4, 295308.
R. W. Robinson, Notes  "A Present for Neil Sloane"
R. W. Robinson, Notes  computer printout


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[[i1]], k+1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[Sum[If[Mod[v[[i]] v[[j]], 2] == 0, GCD[v[[i]], v[[j]]], 0], {j, 1, i  1}], {i, 2, Length[v]}] + Sum[If[Mod[v[[i]], 2] == 0, 2 Quotient[v[[i]], 4], 0], {i, 1, Length[v]}];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p]*2^Length[p], {p, IntegerPartitions[n]}]; s/n!];
Array[a, 19, 0] (* JeanFrançois Alcover, Aug 17 2019, 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[i1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i1, if(v[i]*v[j]%2==0, gcd(v[i], v[j])))) + sum(i=1, #v, if(v[i]%2==0, v[i]\4*2))}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)*2^#p); s/n!} \\ Andrew Howroyd, Sep 25 2018


CROSSREFS

Cf. A004103, A004104, A004105, A004107, A320490.
Sequence in context: A186927 A177010 A300484 * A188498 A012886 A078918
Adjacent sequences: A004103 A004104 A004105 * A004107 A004108 A004109


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


EXTENSIONS

a(0)=1 prepended and a(17)a(18) added by Andrew Howroyd, Sep 25 2018


STATUS

approved



