cp's OEIS Frontend

A004106 Number of line-self-dual nets (or edge-self-dual nets) with n nodes.

%I A004106 M0889 #39 Jul 10 2024 15:02:59
%S A004106 1,2,3,8,29,148,1043,11984,229027,6997682,366204347,30394774084,
%T A004106 4363985982959,994090870519508,393850452332173999,
%U A004106 249278602955869472540,275042591834324901085904,488860279973733024992540668,1514493725905920009795681408275
%N A004106 Number of line-self-dual nets (or edge-self-dual nets) with n nodes.
%C A004106 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
%D A004106 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 A004106 R. W. Robinson, personal communication.
%D A004106 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%D A004106 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A004106 Andrew Howroyd, <a href="/A004106/b004106.txt">Table of n, a(n) for n = 0..50</a> (terms 1..22 from R. W. Robinson)
%H A004106 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 A004106 R. W. Robinson, <a href="/A000666/a000666_2.pdf">Notes - "A Present for Neil Sloane"</a>
%H A004106 R. W. Robinson, <a href="/A004102/a004102_1.pdf">Notes - computer printout</a>
%t A004106 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 A004106 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]}];
%t A004106 a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p]*2^Length[p], {p, IntegerPartitions[n]}]; s/n!];
%t A004106 Array[a, 19, 0] (* _Jean-François Alcover_, Aug 17 2019, after _Andrew Howroyd_ *)
%o A004106 (PARI)
%o A004106 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 A004106 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))}
%o A004106 a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)*2^#p); s/n!} \\ _Andrew Howroyd_, Sep 25 2018
%o A004106 (Python)
%o A004106 from itertools import combinations
%o A004106 from math import prod, gcd, factorial
%o A004106 from fractions import Fraction
%o A004106 from sympy.utilities.iterables import partitions
%o A004106 def A004106(n): return int(sum(Fraction(3**(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2) if not (r&1 and s&1))+sum(((q>>1)&-2)*r+(q*r*(r-1)>>1) for q, r in p.items() if q&1^1))<<s,prod(q**r*factorial(r) for q, r in p.items())) for s, p in partitions(n,size=True))) # _Chai Wah Wu_, Jul 10 2024
%Y A004106 Cf. A004103, A004104, A004105, A004107, A320490.
%K A004106 nonn
%O A004106 0,2
%A A004106 _N. J. A. Sloane_
%E A004106 a(0)=1 prepended and a(17)-a(18) added by _Andrew Howroyd_, Sep 25 2018