A004102 -id:A004102 - cp's OEIS frontend

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

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

A053420 Number of 4-multigraphs on n nodes.

Original entry on oeis.org

1, 5, 35, 900, 90005, 43571400, 95277592625, 925609100039625, 40119721052610123750, 7833164300852979350336250, 6953552738579427778531249187500, 28293472829338822230349054996265275000, 531350037528849507720092485196308155336875000

Offset: 1

Vladeta Jovovic, Jan 11 2000

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.

Andrew Howroyd, Table of n, a(n) for n = 1..50
Harald Fripertinger, The cycle type of the induced action on 2-subsets
Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes

Column k=4 of A063841.

Cf. A000088, A004102, A053400.

from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A053420(n): return int(sum(Fraction(5**(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum((q>>1)*r+(q*r*(r-1)>>1) 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

Terms a(12) and beyond from Andrew Howroyd, Oct 22 2017

A053421 Number of 5-multigraphs on n nodes.

Original entry on oeis.org

1, 6, 56, 2451, 533358, 661452084, 4364646955812, 152397092027960154, 28427450083725134688228, 28645398830642924774967347088, 157458251108667629202718200130101672, 4760428376101385226312810920945121043818096

Offset: 1

Vladeta Jovovic, Jan 11 2000

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.

Andrew Howroyd, Table of n, a(n) for n = 1..50
Harald Fripertinger, The cycle type of the induced action on 2-subsets
Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes

Column k=5 of A063841.

Cf. A000088, A004102, A053400.

from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A053421(n): return int(sum(Fraction(6**(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum((q>>1)*r+(q*r*(r-1)>>1) 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(12) from Andrew Howroyd, Oct 22 2017

A053513 Number of 2-multigraphs with loops on n nodes.

Original entry on oeis.org

3, 18, 165, 3132, 137268, 15548004, 4679446950, 3771927027864, 8186669639820081, 48184182482857319682, 774912347548961791914585, 34299111628183837790980740312, 4205499936656520106909422649497294

Offset: 1

Vladeta Jovovic, Jan 14 2000

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. A000666, A004102, A053514.

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[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2] + 1];
a[n_] := (s=0; Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Array[a, 15] (* Jean-François Alcover, Jul 08 2018, 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, gcd(v[i],v[j]))) + sum(i=1, #v, v[i]\2 + 1)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ Andrew Howroyd, Oct 22 2017

from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A053513(n): return int(sum(Fraction(3**(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum(((q>>1)+1)*r+(q*r*(r-1)>>1) 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

A007127 Definition (1): Number of unlabeled strength-2 Eulerian graphs with n nodes.

Original entry on oeis.org

1, 2, 5, 18, 100, 1242, 43425, 4925635, 1678993887, 1613875721946, 4293014800909806, 31574944534364259507, 644483327087699659771857, 36676558984788056550610362834, 5846161177591490590945591554686844, 2621219060849255874034814155021919844156

Offset: 1

N. J. A. Sloane

Definition (2): Number of Eulerian 2-multigraphs with n nodes.
Definition (3): Number of switching classes of signed graphs on n unlabeled nodes.
Definition (4): Number of switching classes of 2-multigraphs with n nodes
Definition (1) is same as Definition (2). - Vladeta Jovovic, Mar 15 2009
Definition (3) is same as Definition (4). - Vladeta Jovovic, Mar 15 2009
That Definition (2) is same as Definition (4) follows from Theorem 8.3 of Cameron (1977). - N. J. A. Sloane, Mar 22 2009

F. C. Bussemaker, P. J. Cameron, J. J. Seidel, and S. V. Tsaranov, Tables of signed graphs, Report-WSK 91-01, Eindhoven University of Technology, Department of Mathematics and Computing Science, Eindhoven, 1991, 105 pp. (MR: 92g:05001).
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1980.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. C. Bussemaker, P. J. Cameron, J. J. Seidel, and S. V. Tsaranov, Tables of signed graphs, Report-WSK 91-01, Eindhoven University of Technology, Department of Mathematics and Computing Science, Eindhoven, 1991, 105 pp. (MR: 92g:05001).
P. J. Cameron, Cohomological aspects of two-graphs, Math. Zeit., 157 (1977), 101-119.

Cf. A002854, A084565, A004102.

Extended by N. J. A. Sloane from Robinson's table, Oct 20 2006
Edited by N. J. A. Sloane, Mar 30 2009. Does the Cameron paper provide a formula? What about the labeled version? (Compare A002854.)
Corrected by Vladeta Jovovic, Apr 04 2009

A034892 Number of balanced signed graphs on n unlabeled nodes.

Original entry on oeis.org

1, 1, 3, 8, 39, 226, 2283, 36789, 1062679, 55717077, 5405078682, 972656526492, 325183692812200, 202373967993972497, 235081289816026793049, 511296223391186047847309, 2088912833728676472658628201, 16081914207958884651686215477871, 234010862353438997655954463710225233

Offset: 0

Ronald C. Read

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Wikipedia, Signed graph

Cf. A004102, A005142, A318590.

Euler transform of A318590.

Name clarified and offset corrected by Andrew Howroyd, Sep 25 2018

a(0)=1 prepended and terms a(13) and beyond from Andrew Howroyd, Sep 25 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

A199840 Triangle read by rows: T(n,k) is the number of 2-multigraphs on n nodes having exactly k edges, with n >= 1 and 0 <= k <= n*(n-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 5, 8, 9, 12, 9, 8, 5, 3, 1, 1, 1, 1, 3, 6, 14, 24, 43, 62, 87, 100, 110, 100, 87, 62, 43, 24, 14, 6, 3, 1, 1, 1, 1, 3, 7, 18, 40, 91, 180, 352, 616, 1006, 1483, 2036, 2522, 2891, 3012, 2891, 2522, 2036, 1483, 1006, 616, 352, 180, 91, 40, 18, 7, 3, 1, 1

Offset: 1

Geoffrey Critzer, Nov 11 2011

Here a 2-multigraph is an unlabeled graph with at most 2 edges connecting any vertex pair with no self loops allowed.

			Triangle begins:
  1;
  1, 1, 1;
  1, 1, 2, 2, 2, 1, 1;
  1, 1, 3, 5, 8, 9, 12, 9, 8, 5, 3, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (first 20 rows)

Row sums are A004102.

Cf. A008406.

Table[CoefficientList[Expand[PairGroupIndex[SymmetricGroup[n],s] /. Table[s[i]->1+x^i+x^(2i), {i,1,Binomial[n,2]}]], x], {n, 1, 6}]
(* Second program: *)
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_, t_] := Product[g = GCD[v[[i]], v[[j]] ]; t[v[[i]]*v[[j]]/g]^g, {i, 2, Length[v]}, {j, 1, i - 1}]*Product[c = v[[i]]; t[c]^Quotient[c - 1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
row[n_] := Module[{s=0}, Do[s += permcount[p]*edges[p, 1 + x^# + x^(2#)&], {p, IntegerPartitions[n]}]; s/n!] // Expand // CoefficientList[#, x]&;
Array[row, 6] // Flatten (* Jean-François Alcover, Jan 08 2021, 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, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
Row(n) = {my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i+(x^2)^i)); Vecrev(s/n!)}
{ for(n=1, 6, print(Row(n))) } \\ Andrew Howroyd, Nov 07 2019

Terms a(46) and beyond from Andrew Howroyd, Nov 07 2019

A320488 Inverse Euler transform of A004104.

Original entry on oeis.org

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

cp's OEIS Frontend

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

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A053420 Number of 4-multigraphs on n nodes.

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A053421 Number of 5-multigraphs on n nodes.

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A053513 Number of 2-multigraphs with loops on n nodes.

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A007127 Definition (1): Number of unlabeled strength-2 Eulerian graphs with n nodes.

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A034892 Number of balanced signed graphs on n unlabeled nodes.

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

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A199840 Triangle read by rows: T(n,k) is the number of 2-multigraphs on n nodes having exactly k edges, with n >= 1 and 0 <= k <= n*(n-1).

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