A004103 -id:A004103 - cp's OEIS frontend

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

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

A320489 Number of connected nets on n unlabeled nodes.

Original entry on oeis.org

1, 2, 6, 40, 581, 17954, 1376484, 275524246, 147627941386, 213261119287108, 836129127802137203, 8961301472286672235092, 264386586050074867407985650, 21609889100291959988235675761782, 4921292113269596022145447106163528469, 3138303995382684496665239674427115214538716

Offset: 0

N. J. A. Sloane, Oct 25 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50
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. 274.
N. J. A. Sloane, Transforms

Cf. A004103.

EULERi transform of A004103.

a(0)=1 prepended by Andrew Howroyd, Jan 26 2020

A320491 Number of connected edge-self-dual nets with n nodes.

Original entry on oeis.org

1, 2, 0, 4, 13, 98, 748, 9982, 205290, 6546960, 352371887, 29668304472, 4303531187466, 985391751078694, 391866509275972993, 248491878411568471280, 274544424465100159954078, 488310442484985365942048784, 1513516279388226593561422201905

Offset: 0

N. J. A. Sloane, Oct 25 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. 275.
Andrew Howroyd, PARI Program

Cf. A004103 (nets), A004106 (edge-self-dual), A320489 (connected nets).

\\ See link for program.
A320491seq(20) \\ Andrew Howroyd, Jan 28 2020

a(2*n-1) = b(2*n-1), a(2*n) = b(2*n) - (A320489(n) - a(n))/2 where b is the Inverse Euler transform of A004106.

Dead sequence restored, corrected and extended by Andrew Howroyd, Jan 28 2020

A320994 Number of connected point-self-dual nets with 2n nodes.

Original entry on oeis.org

1, 2, 37, 3264, 1798306, 7066174625, 208496688495494, 47481277563116098111, 85161165189313899034899294, 1221965295353715648352925546245057, 142024241427456183309163988600775633635361, 135056692113925953789612785828652550808044930178235

Offset: 0

N. J. A. Sloane, Oct 26 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..40
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. 274.
Andrew Howroyd, PARI Program

Cf. A004103 (nets), A004105 (point-self-dual on 2n nodes), A320489 (connected nets).

\\ See link for program.
A320994seq(15) \\ Andrew Howroyd, Jan 27 2020

a(2*n-1) = b(2*n-1) - A320489(2*n-1)/2, a(2*n) = b(2*n) - (A320489(2*n)-a(n))/2 where b is the Inverse Euler transform of A004105. - Andrew Howroyd, Jan 27 2020

a(0)=1 prepended and terms a(7) and beyond from Andrew Howroyd, Jan 26 2020

A320995 Number of connected self-dual nets with 2n nodes.

Original entry on oeis.org

1, 0, 5, 136, 24162, 29488085, 286615837574, 21717610066598371, 12980514969049888065118, 62082684164458190567999459967, 2405195234525224724112302276711929089, 762399076229936058613587754015434541854738381

Offset: 0

N. J. A. Sloane, Oct 26 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..40
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. 275.
Andrew Howroyd, PARI Program

Cf. A004103 (not necessarily connected nets), A004107 (self-dual), A320489 (connected nets).

\\ See link for program.
A320995seq(15) \\ Andrew Howroyd, Jan 27 2020

a(2*n-1) = b(2*n-1) - A320489(2*n-1)/2, a(2*n) = b(2*n) - (A320489(2*n)-a(n))/2 where b is the Inverse Euler transform of A004107. - Andrew Howroyd, Jan 27 2020

a(0)=1 prepended and terms a(7) and beyond from Andrew Howroyd, Jan 26 2020

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A320489 Number of connected nets on n unlabeled nodes.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A320491 Number of connected edge-self-dual nets with n nodes.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions

A320994 Number of connected point-self-dual nets with 2n nodes.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions

A320995 Number of connected self-dual nets with 2n nodes.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions