A053516 -id:A053516 - cp's OEIS frontend

A004105 Number of point-self-dual nets with 2n nodes. Also number of directed 2-multigraphs with loops on n nodes.

1, 3, 45, 3411, 1809459, 7071729867, 208517974495911, 47481903377454219975, 85161307642554753639601848, 1221965550839348597865127102714827, 142024245093355901785105779901319683262778, 135056692539998733060710198802224149631056479068139

Offset: 0

N. J. A. Sloane

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).

Also nonisomorphic relations on 3-state logic.

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).

R. W. Robinson and Alois P. Heinz, Table of n, a(n) for n = 0..40 (terms n = 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

Cf. A000595, A001374, A053467, A053516.

Prepend[Table[CycleIndex[Join[PairGroup[SymmetricGroup[n],Ordered], Permutations[Range[n^2-n+1,n^2]],2],s]/.Table[s[i]->3,{i,1,n^2-n}],{n,2,7}],1] (* Geoffrey Critzer, Oct 20 2012 *)
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[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v];
a[n_] := (s=0; Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Array[a, 15, 0] (* 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, 2*gcd(v[i],v[j]))) + sum(i=1, #v, v[i])}
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 A004105(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*r**2 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 10 2024

a(n) = Sum_{1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/ (1^s_1*s_1!*2^s_2*s_2!*...)) where fixA[s_1, s_2, ...] = 3^Sum_{i, j>=1} (gcd(i,j)*s_i*s_j).

More terms from Vladeta Jovovic, Jan 14 2000

Formula from Christian G. Bower, Jan 06 2004

A001374 Number of relational systems on n nodes. Also number of directed 3-multigraphs with loops on n nodes.

Original entry on oeis.org

4, 136, 44224, 179228736, 9383939974144, 6558936236286040064, 62879572771326489528942592, 8439543710699844562674685252214784, 16110027001555070629022725866559372785352704, 442829046878106126159584032189649757399796014050181120

Offset: 1

N. J. A. Sloane

W. Oberschelp, "Strukturzahlen in endlichen Relationssystemen", in Contributions to Mathematical Logic (Proceedings 1966 Hanover Colloquium), pp. 199-213, North-Holland Publ., Amsterdam, 1968.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 1..40
W. Oberschelp, Strukturzahlen in endlichen Relationssystemen, in Contributions to Mathematical Logic (Proceedings 1966 Hanover Colloquium), pp. 199-213, North-Holland Publ., Amsterdam, 1968. [Annotated scanned copy]

Cf. A000595, A004105, A053468, A053516.

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[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v];
a[n_] := (s=0; Do[s += permcount[p]*4^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, 2*gcd(v[i],v[j]))) + sum(i=1, #v, v[i])}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*4^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 A001374(n): return int(sum(Fraction(1<<((sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))<<1)+sum(q*r**2 for q, r in p.items())<<1),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 10 2024

More terms from Vladeta Jovovic, Jan 14 2000

A052114 Number of self-complementary directed 4-multigraphs with loops on n nodes.

Original entry on oeis.org

1, 13, 313, 52891, 30554141, 89011081055, 1243751028948305, 70334570607769968970, 23735285427941643311618345, 27755772992017058140287194221448, 226477787759401129853705684271059207073, 5698414152656591747538959168064394745850705494

Offset: 1

Vladeta Jovovic, Jan 21 2000

nonn

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

Cf. A052113, A053516, A053588.

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, if(v[i]*v[j]%2==0, gcd(v[i],v[j])))) + sum(i=1, #v, if(v[i]%2==0, v[i]))}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*5^edges(p)); s/n!} \\ Andrew Howroyd, Sep 17 2018

Terms a(11) and beyond from Andrew Howroyd, Sep 17 2018

A154238 Number of orbits of the action g*b = b o (g x g) of the group of permutations g of an n-element set S on the set of closed binary operations b on S.

Original entry on oeis.org

1, 1, 10, 3411, 179228736, 2483590604688125, 14325593551925794051596768, 50976900379139614139041610902600299311, 155682086692129060007763454017522652304844346252853248

Offset: 0

David Pasino, Jan 05 2009, Jan 08 2009, Jan 12 2009

nonn

Here are several different ways of expressing the condition that g*b = b:
b(u, v) = b(gu, gv) for all u, v in S.
The level sets of b are closed under g x g.
The level sets of b are unions of cycles of g x g.
The cycles of g x g are subsets of level sets of b.
b is constant on cycles of g x g.
There is no requirement for g to be an automorphism of b. Given g, the fixed b are determined by simply choosing a value in S for each cycle of g x g. The product b(u, v) is defined to be that constant value for every (u, v) in the cycle.
So the number of degrees of freedom for b is the number of cycles of g x g. How many cycles does g have on S x S? If u is in a c-cycle C and v is in a d-cycle D, then (u, v) is in an lcm(c, d)-cycle and C x D is partitioned into these cycles, so there must be cd/lcm(c, d) of them, which is gcd(c, d).
So letting s_k be the number of k-cycles of g on S for each k from 1 to n, the total number of cycles of g on S x S is the sum on k and j of gcd(k, j) s_k s_j. That's the number of degrees of freedom for b and each degree has valence n, so raise n to that power. Then multiply by the well-known number of permutations of type s, which is n! divided by the factorials of the s_k and by the powers k^s_k. Add this up over all the partitions of n and divide by n!.
Additional comments from Christian G. Bower: This is the number of nonisomorphic n-state relations on a set of n elements. If at the step of raising n to the power, we raised instead some constant m to that power, the formula would give the number of isomorphism classes of m-state relations on an n-element set.

Alois P. Heinz, Table of n, a(n) for n = 0..26

Cf. k-state relations: A000595 for k=2, A004105 for k=3, A001374 for k=4, A053516 for k=5.
Cf. A001329, A091057.
Cf. A000595, A004105, A001374, A053516. - Vladeta Jovovic, Jan 06 2009

a(n) = Sum_{1*s_1 + 2*s_2 + ... = n} (fixA[s_1, s_2,..]/(1^s_1*s_1!*2^s_2*s2!* ...)) where fixA[s_1, s_2, ...] = n^(Sum_{i, j>=1} gcd(i, j)*s_i*s_j).

Edited by Christian G. Bower and N. J. A. Sloane, Jan 08 2009

cp's OEIS Frontend

A004105 Number of point-self-dual nets with 2n nodes. Also number of directed 2-multigraphs with loops on n nodes.

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A001374 Number of relational systems on n nodes. Also number of directed 3-multigraphs with loops on n nodes.

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A052114 Number of self-complementary directed 4-multigraphs with loops on n nodes.

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A154238 Number of orbits of the action g*b = b o (g x g) of the group of permutations g of an n-element set S on the set of closed binary operations b on S.

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