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 1-6 of 6 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

A003028 Number of digraphs on n labeled nodes with a source.

Original entry on oeis.org

1, 3, 51, 3614, 991930, 1051469032, 4366988803688, 71895397383029040, 4719082081411731363408, 1237678715644664931691596416, 1297992266840866792981316221144960, 5444416466164313011147841248189209354496, 91343356480627224177654291875698256656613808896
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Here a source is a node that is connected by a directed path to every other node in the digraph (but does not necessarily have indegree zero). - Geoffrey Critzer, Apr 14 2023

References

  • V. Jovovic and 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 A051421.
Row sums of A057274.
Column k=1 of A361579.
Cf. A003027, A003029, A003030, A049524, A049414, A350792.

Programs

Extensions

Corrected and extended by Vladeta Jovovic, Goran Kilibarda
Terms a(12) and beyond from Andrew Howroyd, Jan 11 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

A057277 Triangle T(n,k) of number of digraphs with a source on n unlabeled nodes with k arcs, k=0..n*(n-1).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 2, 4, 4, 1, 1, 0, 0, 0, 4, 16, 34, 46, 38, 27, 13, 5, 1, 1, 0, 0, 0, 0, 9, 56, 229, 573, 1058, 1448, 1653, 1487, 1153, 707, 379, 154, 61, 16, 5, 1, 1, 0, 0, 0, 0, 0, 20, 198, 1218, 5089, 15596, 37302, 72776, 119531, 168233, 205923, 220337, 207147, 170965, 124099, 78811, 43861, 21209, 8951, 3242, 1043, 288, 76, 17, 5, 1, 1
Offset: 1

Views

Author

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

Keywords

Examples

			Triangle begins:
  [1],
  [0, 1, 1],
  [0, 0, 2, 4, 4, 1, 1],
  [0, 0, 0, 4, 16, 34, 46, 38, 27, 13, 5, 1, 1],
  ....
The number of digraphs with a source on 3 unlabeled nodes is 12 = 2+4+4+1+1.

Links

Crossrefs

Row sums give A051421.
Column sums give A350907.
The labeled version is A057274.
Cf. A057270, A057276, A057278, A057279, A350794, A350797.

Programs

Extensions

Terms a(46) and beyond from Andrew Howroyd, Jan 21 2022

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

Views

Author

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    \\ 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

A350793 Triangle read by rows: T(n,k) is the number of digraphs on n labeled nodes with k arcs and a global source (or sink), n >= 1, k = 0..(n-1)^2.

Original entry on oeis.org

1, 0, 2, 0, 0, 9, 12, 3, 0, 0, 0, 64, 252, 396, 320, 144, 36, 4, 0, 0, 0, 0, 625, 4860, 17060, 35900, 50775, 51300, 38340, 21540, 9075, 2800, 600, 80, 5, 0, 0, 0, 0, 0, 7776, 99720, 603720, 2300310, 6206730, 12654384, 20310840, 26385240, 28273620, 25302960
Offset: 1

Views

Author

Andrew Howroyd, Jan 17 2022

Keywords

Examples

			Triangle begins:
  [1] 1;
  [2] 0, 2;
  [3] 0, 0, 9, 12, 3;
  [4] 0, 0, 0, 64, 252, 396, 320, 144, 36, 4;
  ...

Links

Crossrefs

Row sums are A350792.
The leading diagonal is A000169.
The unlabeled version is A350797.
Cf. A057274, A350791.

Programs

  • PARI
    InitiallyV(n, e=2)={my(v=vector(n)); for(n=1, n, v[n] = n*e^((n-1)^2) - sum(k=1, n-1, binomial(n,k)*e^((n-2)*(n-k))*v[k])); v}
row(n)={Vecrev(InitiallyV(n, 1+'y)[n])}
{ for(n=1, 5, print(row(n))) }
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.