A102596 -id:A102596 - cp's OEIS frontend

A369283 Triangle read by rows: T(n,k) is the number of labeled point-determining graphs with n nodes and k edges, n >= 0, 0 <= k <= n*(n - 1)/2.

1, 1, 0, 1, 0, 3, 0, 1, 0, 0, 3, 16, 12, 0, 1, 0, 0, 15, 60, 130, 132, 140, 80, 30, 0, 1, 0, 0, 0, 15, 600, 1692, 3160, 4635, 4620, 3480, 2088, 885, 240, 60, 0, 1, 0, 0, 0, 105, 1260, 7665, 28042, 74280, 142380, 218960, 271404, 276150, 230860, 157710, 86250, 38752, 13524, 3360, 560, 105, 0, 1

Offset: 0

Andrew Howroyd, Jan 18 2024

Point-determining graphs are also called mating graphs.

			Triangle begins:
 [0] 1;
 [1] 1;
 [2] 0, 1;
 [3] 0, 3,  0,  1;
 [4] 0, 0,  3, 16,  12,    0,    1;
 [5] 0, 0, 15, 60, 130,  132,  140,   80,   30,    0,    1;
 [6] 0, 0,  0, 15, 600, 1692, 3160, 4635, 4620, 3480, 2088, 885, 240, 60, 0, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1350 (rows 0..20)
Ira M. Gessel and Ji Li, Enumeration of point-determining graphs, J. Combinatorial Theory Ser. A 118 (2011), 591-612.

Row sums are A006024.

Cf. A102579, A102596, A368987 (unlabeled).

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(p,t) = {prod(i=2, #p, prod(j=1, i-1, t(p[i]*p[j])))}
row(n) = {my(s=0); forpart(p=n, s += permcount(p)*(-1)^(n-#p)*edges(p, w->1 + x^w)); Vecrev(s)}

Sum_{k>=0} 2^k*T(n,k) = A102596(n).

Sum_{k>=0} 3^k*T(n,k) = A102579(n).

A102579 Number of n-node labeled digraphs whose underlying graphs are mating graphs, cf. A006024.

Original entry on oeis.org

1, 1, 3, 36, 2160, 577260, 629332740, 2778611237640, 49231195558057800, 3472536190055702938560, 971496134741368475512176960, 1076456769176854177692230640971520, 4723714896453453856832035698858163891200, 82155195331101404797144020808246647196388279040

Offset: 0

Goran Kilibarda, Zoran Maksimovic, Vladeta Jovovic, Jan 21 2005

nonn

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

Cf. A006024, A102580 (connected), A102596 (oriented), A102598, A369283.

\\ permcount, edges defined in A369283.
a(n) = {my(s=0); forpart(p=n, s += permcount(p)*(-1)^(n-#p)*edges(p, w->1 + 3^w)); s} \\ Andrew Howroyd, Jan 20 2024

a(n) = Sum_{k>=0} 3^k * A369283(n,k). - Andrew Howroyd, Jan 20 2024

a(0)=1 prepended and a(12) onwards from Andrew Howroyd, Jan 20 2024

A102597 Number of n-node labeled connected oriented graphs whose underlying graphs are mating graphs, cf. A006024.

Original entry on oeis.org

1, 1, 2, 8, 352, 32768, 9072896, 7389698048, 17695056717824, 124756911516516352, 2594584522679818518528, 159570269132472101976145920, 29103249711329178773313159168000, 15783921191009840333055992641622114304, 25514864105378775907711095288995558725779456

Offset: 0

Goran Kilibarda, Zoran Maksimovic, Vladeta Jovovic, Jan 22 2005

nonn

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

Cf. A006024, A102580 (digraphs), A102596 (not necessarily connected), A102599 (unlabeled), A369283.

\\ b(n) is A102596(n); permcount, edges defined in A369283.
b(n) = {my(s=0); forpart(p=n, s += permcount(p)*(-1)^(n-#p)*edges(p, w->1 + 2^w)); s}
seq(n) = {Vec(serlaplace(1 + x + log(sum(k=0, n, b(k)*x^k/k!, O(x*x^n))/(1 + x))))} \\ Andrew Howroyd, Jan 20 2024

E.g.f.: 1 + x + log(B(x)/(1 + x)) where B(x) is the e.g.f. of A102596. - Andrew Howroyd, Jan 20 2024

a(0)=1 prepended and a(13) onwards from Andrew Howroyd, Jan 20 2024

cp's OEIS Frontend

A369283 Triangle read by rows: T(n,k) is the number of labeled point-determining graphs with n nodes and k edges, n >= 0, 0 <= k <= n*(n - 1)/2.

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A102579 Number of n-node labeled digraphs whose underlying graphs are mating graphs, cf. A006024.

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A102597 Number of n-node labeled connected oriented graphs whose underlying graphs are mating graphs, cf. A006024.

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