A057272 -id:A057272 - cp's OEIS frontend

A057273 Triangle T(n,k) of the number of strongly connected digraphs on n labeled nodes and with k arcs, k=0..n*(n-1).

1, 0, 0, 1, 0, 0, 0, 2, 9, 6, 1, 0, 0, 0, 0, 6, 84, 316, 492, 417, 212, 66, 12, 1, 0, 0, 0, 0, 0, 24, 720, 6440, 26875, 65280, 105566, 122580, 106825, 71700, 37540, 15344, 4835, 1140, 190, 20, 1, 0, 0, 0, 0, 0, 0, 120, 6480, 107850, 868830, 4188696, 13715940

Offset: 1

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

			Triangle begins:
  [1] 1;
  [2] 0,0,1;
  [3] 0,0,0,2,9,6,1;
  [4] 0,0,0,0,6,84,316,492,417,212,66,12,1;
  ...
Number of strongly connected digraphs on 3 labeled nodes is 18 = 2+9+6+1.

Archer, K., Gessel, I. M., Graves, C., & Liang, X. (2020). Counting acyclic and strong digraphs by descents. Discrete Mathematics, 343(11), 112041. See Table 2.

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)
Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, arXiv:1909.01550 [math.CO], 20 Mar 2020. See Table 2.

Row sums give A003030.
The unlabeled version is A057276.
Cf. A057271, A057272, A057274, A057275, A062735.
Cf. A123554, A339590, A339807.

B(nn, e=2)={my(v=vector(nn)); for(n=1, nn, v[n] = e^(n*(n-1)) - sum(k=1, n-1, binomial(n,k)*e^((n-1)*(n-k))*v[k])); v}
Strong(n, e=2)={my(u=B(n, e), v=vector(n)); v[1]=1; for(n=2, #v, v[n] = u[n] + sum(j=1, n-1, binomial(n-1, j-1)*u[n-j]*v[j])); v}
row(n)={ Vecrev(Strong(n, 1+'y)[n]) }
{ for(n=1, 5, print(row(n))) } \\ Andrew Howroyd, Jan 10 2022

Terms a(46) and beyond from Andrew Howroyd, Jan 10 2022

A057271 Triangle T(n,k) of number of digraphs with a source and a sink on n labeled nodes and k arcs, k=0,1,..,n*(n-1).

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 6, 20, 15, 6, 1, 0, 0, 0, 24, 234, 672, 908, 792, 495, 220, 66, 12, 1, 0, 0, 0, 0, 120, 2544, 16880, 55000, 111225, 161660, 183006, 167660, 125945, 77520, 38760, 15504, 4845, 1140, 190, 20, 1

Offset: 1

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

			Triangle starts:
[1] 1;
[2] 0,2,1;
[3] 0,0,6,20,15,6,1;
[4] 0,0,0,24,234,672,908,792,495,220,66,12,1;
  ...
The number of digraphs with a source and a sink on 3 labeled nodes is 48 = 6+20+15+6+1.

V. Jovovic, G. Kilibarda, Enumeration of labeled initially-finally connected digraphs, Scientific review, Serbian Scientific Society, 19-20 (1996), p. 245.

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)
V. Jovovic and G. Kilibarda, Enumeration of labeled quasi-initially connected digraphs, Discrete Math., 224 (2000), 151-163.
R. W. Robinson, Counting digraphs with restrictions on the strong components, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.

Row sums give A049524.
The unlabeled version is A057278.
Cf. A057272, A057273, A057274, A057275, A062735, A350791.

\\ Following Eqn 20 in the Robinson reference.
Z(p,f)={my(n=serprec(p,x)); serconvol(p, sum(k=0, n-1, x^k*f(k), O(x^n)))}
G(e,p)={Z(p, k->1/e^(k*(k-1)/2))}
U(e,p)={Z(p, k->e^(k*(k-1)/2))}
DigraphEgf(n,e)={sum(k=0, n, e^(k*(k-1))*x^k/k!, O(x*x^n) )}
StrongD(n,e=2)={-log(U(e, 1/G(e, DigraphEgf(n, e))))}
InitFinally(n, e=2)={my(S=StrongD(n, e)); Vec(serlaplace( S - S^2 + exp(S) * U(e, G(e, S*exp(-S))^2*G(e, DigraphEgf(n,e))) ))}
row(n)={Vecrev(InitFinally(n, 1+'y)[n]) }
{ for(n=1, 5, print(row(n))) } \\ Andrew Howroyd, Jan 16 2022

A057274 Triangle T(n,k) of the number of digraphs with a source on n labeled nodes with k arcs, k=0..n*(n-1).

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 9, 20, 15, 6, 1, 0, 0, 0, 64, 330, 720, 914, 792, 495, 220, 66, 12, 1, 0, 0, 0, 0, 625, 5804, 24560, 63940, 117310, 164260, 183716, 167780, 125955, 77520, 38760, 15504, 4845, 1140, 190, 20, 1

Offset: 1

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

			Triangle begins:
  1;
  0, 2, 1;
  0, 0, 9, 20,  15,   6,   1;
  0, 0, 0, 64, 330, 720, 914, 792, 495, 220, 66, 12, 1;
  ...
The number of digraphs with a source on 3 labeled nodes is the sum of the terms in row 3, i.e., 0+0+9+20+15+6+1 = 51 = A003028(3).

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)
V. Jovovic and G. Kilibarda, Enumeration of labeled quasi-initially connected digraphs, Discrete Math., 224 (2000), 151-163.

Row sums give A003028.
The unlabeled version is A057277.
Cf. A057271, A057272, A057273, A057275, A062735.

\\ See A057273 for Strong.
Lambda(t, nn, e=2)={my(v=vector(1+nn)); for(n=0, nn, v[1+n] = e^(n*(n+t-1)) - sum(k=0, n-1, binomial(n,k)*e^((n-1)*(n-k))*v[1+k])); v}
Initially(n, e=2)={my(s=Strong(n, e), v=vector(n)); for(k=1, n, my(u=Lambda(k, n-k, e)); for(i=k, n, v[i] += binomial(i,k)*u[1+i-k]*s[k])); v }
row(n)={ Vecrev(Initially(n, 1+'y)[n]) }
{ for(n=1, 5, print(row(n))) } \\ Andrew Howroyd, Jan 11 2022

A049414 Number of quasi-initially connected digraphs with n labeled nodes.

Original entry on oeis.org

1, 3, 54, 3804, 1022320, 1065957628, 4389587378792, 72020744942708040, 4721708591209396542528, 1237892622263984613044109216, 1298060581376190776821670648395840

Offset: 1

Vladeta Jovovic, Goran Kilibarda

nonn

We say that a node v of a digraph is a quasi-source iff for every other node u there exists directed path from u to v or from v to u. A digraph with at least one quasi-source is called quasi-initially connected.

Sean A. Irvine, Java program (github)
V. Jovovic and G. Kilibarda, Enumeration of labeled quasi-initially connected digraphs, Discrete Math., 224 (2000), 151-163.

Cf. A003027, A003028, A003029, A003030.

Row sums of A057272.

The recurrence formulas are too long to be presented here.

A057275 Triangle T(n,k) of number of unilaterally connected digraphs on n labeled nodes and with k arcs, k=0..n*(n-1).

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 6, 20, 15, 6, 1, 0, 0, 0, 24, 222, 660, 908, 792, 495, 220, 66, 12, 1, 0, 0, 0, 0, 120, 2304, 15540, 52700, 109545, 161120, 182946, 167660, 125945, 77520, 38760, 15504, 4845, 1140, 190, 20, 1

Offset: 1

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

			Triangle begins:
  [1],
  [0,2,1],
  [0,0,6,20,15,6,1],
  [0,0,0,0,24,222,660,908,792,495,220,66,12,1],
  ...
The number of unilaterally connected digraphs on 3 labeled nodes is 48 = 6+20+15+6+1.

Andrew Howroyd, Table of n, a(n) for n = 1..2680
V. Jovovic and G. Kilibarda, Enumeration of labeled quasi-initially connected digraphs, Discrete Math., 224 (2000), 151-163.

Row sums give A003029.
The unlabeled version is A057270.
Cf. A057271, A057272, A057273, A057274, A062735.

\\ See A057273 for Strong.
Unilaterally(n, e=2)={my(u=vector(n), s=Strong(n,e)); for(n=1, #u, u[n]=vector(n, k, binomial(n,k)*s[k]*if(k==n, 1, sum(j=1, n-k, e^(k*(n-k-j))*(e^(k*j)-1)*u[n-k][j])))); vector(#u, n, vecsum(u[n]))}
row(n)={Vecrev(Unilaterally(n, 1+'y)[n])}
{ for(n=1, 5, print(row(n))) } \\ Andrew Howroyd, Jan 19 2022

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A057273 Triangle T(n,k) of the number of strongly connected digraphs on n labeled nodes and with k arcs, k=0..n*(n-1).

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A057271 Triangle T(n,k) of number of digraphs with a source and a sink on n labeled nodes and k arcs, k=0,1,..,n*(n-1).

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A057274 Triangle T(n,k) of the number of digraphs with a source on n labeled nodes with k arcs, k=0..n*(n-1).

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A049414 Number of quasi-initially connected digraphs with n labeled nodes.

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A057275 Triangle T(n,k) of number of unilaterally connected digraphs on n labeled nodes and with k arcs, k=0..n*(n-1).

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