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A004107 Number of self-dual nets with 2n nodes.
(Formerly M4663)
1, 1, 9, 165, 24651, 29522961, 286646256675, 21717897090413481, 12980536689318626076840, 62082697145168772833294318409, 2405195296608025717214293025492960466, 762399078635131851885116768114137369439908725 (list; graph; refs; listen; history; text; internal format)



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


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).


Andrew Howroyd, Table of n, a(n) for n = 0..40 (terms 1..13 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


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_] := 2 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[2 Quotient[v[[i]], 2], {i, 1, Length[v]}];

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

Array[a, 12, 0] (* Jean-Fran├žois Alcover, Aug 17 2019, after Andrew Howroyd *)



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

a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ Andrew Howroyd, Sep 25 2018


Cf. A004103, A004104, A004105, A004106.

Sequence in context: A053130 A219074 A166180 * A180831 A180819 A132874

Adjacent sequences:  A004104 A004105 A004106 * A004108 A004109 A004110




N. J. A. Sloane


a(0)=1 prepended by Andrew Howroyd, Sep 25 2018



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Last modified May 9 17:48 EDT 2021. Contains 343742 sequences. (Running on oeis4.)