A052112
Number of self-complementary directed 2-multigraphs on n nodes.
Original entry on oeis.org
1, 2, 14, 159, 7629, 599456, 226066304, 139178815861, 410179495378288, 2055126126323159298, 48234291396964332998082, 2016523952125103590736221923, 382812826011951187177138562992638, 135681830960694827549160289095792266106
Offset: 1
-
permcount[v_List] := 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_List] := 2 Sum[Sum[If[EvenQ[v[[i]] v[[j]]], GCD[v[[i]], v[[j]]], 0], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[If[EvenQ[v[[i]]], v[[i]] - 1, 0], {i, 1, Length[v]}];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!];
Array[a, 25] (* Jean-François Alcover, Sep 12 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, if(v[i]*v[j]%2==0, gcd(v[i],v[j])))) + sum(i=1, #v, if(v[i]%2==0, v[i]-1))}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ Andrew Howroyd, Sep 16 2018
A052107
Number of self-complementary 3-multigraphs on n nodes.
Original entry on oeis.org
1, 0, 0, 4, 16, 0, 0, 2080, 32896, 0, 0, 178990080, 11453771776, 0, 0, 3002404080455680, 768614611824951296, 0, 0, 10316167090130469587779584, 10563755026498136326181748736, 0, 0, 7561830376433501721102295492903043072, 30973257220603971305905396442627825467392
Offset: 1
- 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.
-
permcount[v_List] := 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_List] := 4 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + 2 Total[v];
a[n_] := Module[{s = 0}, If[Mod[n, 4] < 2, Do[s += permcount[4*p]* 4^edges[p]*If[OddQ[n], n*4^Length[p], 1], {p, IntegerPartitions[ Quotient[n, 4]]}]]; s/n!];
Array[a, 25] (* Jean-François Alcover, Sep 12 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) = {4*sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, 2*v[i])}
a(n) = {my(s=0); if(n%4<2, forpart(p=n\4, s+=permcount(4*Vec(p))*4^edges(p)*if(n%2, n*4^#p, 1))); s/n!} \\ Andrew Howroyd, Sep 17 2018
A052108
Number of self-complementary 5-multigraphs on n nodes.
Original entry on oeis.org
1, 0, 0, 9, 54, 0, 0, 52650, 1890540, 0, 0, 264480947280, 57127543673760, 0, 0, 1295355012667626301200, 1678780080964997690732640, 0, 0, 6577809875294796334824189267538944
Offset: 1
- 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.
-
permcount[v_List] := 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_List] := 4 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + 2 Total[v];
a[n_] := Module[{s = 0}, If[Mod[n, 4] < 2, Do[s += permcount[4*p]* 6^edges[p]*If[OddQ[n], n*6^Length[p], 1], {p, IntegerPartitions[ Quotient[n, 4]]}]]; s/n!];
Array[a, 25] (* Jean-François Alcover, Sep 12 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) = {4*sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, 2*v[i])}
a(n) = {my(s=0); if(n%4<2, forpart(p=n\4, s+=permcount(4*Vec(p))*6^edges(p)*if(n%2, n*6^#p, 1))); s/n!} \\ Andrew Howroyd, Sep 17 2018
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
-
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
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