cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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).

Original entry on oeis.org

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

Views

Author

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

Keywords

Examples

			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.

References

  • 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.

Links

Crossrefs

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

Programs

  • PARI
    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

Extensions

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

Views

Author

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

Keywords

Examples

			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.

References

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

Links

Crossrefs

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

Programs

  • PARI
    \\ 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

Views

Author

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

Keywords

Examples

			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).

Links

Crossrefs

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

Programs

  • PARI
    \\ 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

A003029 Number of unilaterally connected digraphs with n labeled nodes.

Original entry on oeis.org

1, 3, 48, 3400, 955860, 1034141596, 4340689156440, 71756043154026904, 4716284688998245793376, 1237457313794197125403483936, 1297922676419612772784598299454784, 5444329780242560941321703550018388707904, 91342929096228825123968637168641318872709897472
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • V. Jovovic, G. Kilibarda, Enumeration of labeled initially-finally connected digraphs, Scientific review, Serbian Scientific Society, 19-20 (1996), 237-247.
  • R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

The unlabeled version is A003088.
Row sums of A057275.
Cf. A003027, A003028, A003030, A049524, A049414.

Programs

Extensions

Corrected and extended by Vladeta Jovovic, Goran Kilibarda
Terms a(12) and beyond from Andrew Howroyd, Jan 11 2022
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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