A004104 -id:A004104 - cp's OEIS frontend

A320488 Inverse Euler transform of A004104.

1, 1, 0, 1, 4, 14, 65, 572, 7434, 163284, 5736792, 342169618, 33534958026, 5442700283638, 1484664947481018, 664513607618098252, 508538464299684269212, 635542752091150346032474, 1374528064543284187245552390, 4842758246111267151697826493193, 29772724415959420224886585241636839

Offset: 0

N. J. A. Sloane, Oct 25 2018

nonn

The inverse Euler transform of A004104 does not give the number of connected self-dual signed graphs. The combinatorial interpretation of this sequence is that of either a connected self-dual signed graph or a pair of distinct connected signed graphs which are dual to each other (but not self-dual). - Andrew Howroyd, Jan 26 2020

Andrew Howroyd, Table of n, a(n) for n = 0..50
N. J. A. Sloane, Transforms

Cf. A004102, A004104, A053465, A320499.

Definition edited by Andrew Howroyd, Jan 26 2020

A053588 Number of self-complementary 4-multigraphs on n nodes.

Original entry on oeis.org

1, 1, 3, 16, 121, 1480, 50993, 3279685, 505641590, 152461906778, 103587671805408, 145528904385412088, 442626996609870050404, 2918362542591139744394993, 40446812392580562094804791143, 1260273961234324967695235253182680, 80686628450087709982052029871655471264

Offset: 1

Vladeta Jovovic, Jan 19 2000

nonn

V. Jovovic, On the number of m-place relations (in Russian), Logiko-algebraicheskie konstruktsii, Tver, 1992, 59-66.
J. Xu, Ch. R. Wang, J. F. Wang, The theory of self-complementary k-multigraphs (in Chinese), Pure Appl. Math. [Chuncui Shuxue yu Yingyong Shuxue] 10 (1994), Special Issue, 18-22.

Andrew Howroyd, Table of n, a(n) for n = 1..50
D. Wille, Enumeration of self-complementary structures, J. Comb. Theory B 25 (1978) 143-150.

Cf. A004102, A004104, A053420.

permcount[v_] := Module[{m = 1, s = 0, t, i, k = 0}, 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[If[EvenQ[v[[i]]*v[[j]]], GCD[v[[i]], v[[j]]], 0], {i, 2, Length[v]}, {j, 1, i - 1}] + Sum[If[EvenQ[v[[i]]], 2 Quotient[v[[i]], 4], 0], {i, 1, Length[v]}];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*5^edges[p], {p, IntegerPartitions[n]}]; s/n!];
a /@ Range[1, 20] (* Jean-François Alcover, Sep 22 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) = {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)*5^edges(p)); s/n!} \\ Andrew Howroyd, Sep 17 2018

Terms a(16) and beyond from Andrew Howroyd, Sep 17 2018

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

Original entry on oeis.org

1, 2, 3, 8, 29, 148, 1043, 11984, 229027, 6997682, 366204347, 30394774084, 4363985982959, 994090870519508, 393850452332173999, 249278602955869472540, 275042591834324901085904, 488860279973733024992540668, 1514493725905920009795681408275

Offset: 0

N. J. A. Sloane

nonn

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

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..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

Cf. A004103, A004104, A004105, A004107, A320490.

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-François 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) = {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

from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
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))<Chai Wah Wu, Jul 10 2024

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

A004107 Number of self-dual nets with 2n nodes.

Original entry on oeis.org

1, 1, 9, 165, 24651, 29522961, 286646256675, 21717897090413481, 12980536689318626076840, 62082697145168772833294318409, 2405195296608025717214293025492960466, 762399078635131851885116768114137369439908725

Offset: 0

N. J. A. Sloane

nonn

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, and 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

Cf. A004103, A004104, A004105, 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];
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-François 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

from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A004107(n): return int(sum(Fraction(3**((sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))<<1)+sum(((q&-2)+q*(r-1))*r for q, r in p.items())),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 09 2024

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

A052111 Number of self-complementary 2-multigraphs with loops on n nodes.

Original entry on oeis.org

1, 2, 5, 24, 120, 956, 13214, 275848, 10613479, 601955190, 63788179593, 9985272721908, 2906903866536978, 1268802939666164781, 1023198355173637429689, 1258181815243248217067175, 2834890911778762731361375215, 9900896274205100008273760895560

Offset: 1

Vladeta Jovovic, Jan 21 2000

nonn

A 2-multigraph is similar to an ordinary graph except there are 0, 1 or 2 edges between any two nodes (self-loops are not allowed).

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A053513, A004104.

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+1))}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ Andrew Howroyd, Sep 16 2018

from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A052111(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)|1)*r+(q*r*(r-1)>>1) for q, r in p.items() if q&1^1)),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 09 2024

Terms a(17) and beyond from Andrew Howroyd, Sep 16 2018

A320499 Number of connected self-dual signed graphs with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 3, 14, 62, 572, 7409, 163284, 5736443, 342169618, 33534945769, 5442700283638, 1484664946343496, 664513607618098252, 508538464299389501337, 635542752091150346032474, 1374528064543283977151585962, 4842758246111267151697826493193

Offset: 0

N. J. A. Sloane, Oct 26 2018

nonn

Edward A. Bender and E. Rodney Canfield, Enumeration of connected invariant graphs, Journal of Combinatorial Theory, Series B 34.3 (1983): 268-278. See p. 273, fourth table.
Andrew Howroyd, PARI Program

Cf. A004102 (signed graphs), A004104 (self-dual), A053465 (connected signed graphs).

\\ See link for program.
A320499seq(20) \\ Andrew Howroyd, Jan 27 2020

a(2*n-1) = b(2*n-1), a(2*n) = b(2*n) - (A053465(n) - a(n))/2 where b is the Inverse Euler transform of A004104. - Andrew Howroyd, Jan 27 2020

Dead sequence restored by Andrew Howroyd, Jan 26 2020

a(0)=1 prepended and terms a(13) and beyond from Andrew Howroyd, Jan 26 2020

cp's OEIS Frontend

A320488 Inverse Euler transform of A004104.

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A053588 Number of self-complementary 4-multigraphs on n nodes.

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

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A004107 Number of self-dual nets with 2n nodes.

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Mathematica

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A052111 Number of self-complementary 2-multigraphs with loops on n nodes.

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Programs

PARI

Python

Extensions

A320499 Number of connected self-dual signed graphs with n nodes.

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PARI

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Extensions