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A004106 Number of line-self-dual nets (or edge-self-dual 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 line-self-dual 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 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.

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, 295-308.

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[[i-1]], 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] (* Jean-Fran├žois 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[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

edges(v) = {sum(i=2, #v, sum(j=1, i-1, 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

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Last modified September 15 16:12 EDT 2019. Contains 327078 sequences. (Running on oeis4.)