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 1-5 of 5 results.

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

A350360 Number of unlabeled digraphs with n nodes containing a global sink (or source).

Original entry on oeis.org

1, 1, 5, 60, 2126, 236560, 86140208, 105190967552, 442114599155408, 6536225731179398016, 345635717436525206325760, 66213119317905480992415271936, 46409685828045501628276172471067136, 119963222885004355352870426935849790038016
Offset: 1

Views

Author

Jim Snyder-Grant, Dec 26 2021

Keywords

Comments

A global sink is a node that has out-degree zero and to which all other nodes have a directed path.
A global source is a node that has in-degree zero and has a directed path to all other nodes. A digraph with a global source, transposed, is a digraph with a global sink.

Examples

			For n=3, 5 digraph edge-sets: (vertex 0 is the single global sink)
  {10,21,20}
  {21,10}
  {21,12,10}
  {21,12,10,20}
  {20,10}

Links

Crossrefs

The labeled version is A350792.
Row sums of A350797.
Cf. A051421, A049531.
Cf. A049512, A350415, A350794, A350798.

Programs

  • PARI
    \\ See PARI link in A350794 for program code.
A350360seq(15) \\ Andrew Howroyd, Jan 21 2022
  • Sage
    # A simple but slow way is to start from all digraphs and filter
# This code can get to n=5
# The linked C code was used to get to n=7
def one_global_sink(g):
    if (g.out_degree().count(0) != 1): return False;
    s = g.out_degree().index(0)
    return [g.distance(v,s) for v in g.vertices()].count(Infinity) == 0
[len([g for g in digraphs(n) if one_global_sink(g)]) for n in (0..5)]

Extensions

Terms a(8) and beyond from Andrew Howroyd, Jan 21 2022

A350790 Number of digraphs on n labeled nodes with a global source and sink.

Original entry on oeis.org

1, 2, 12, 432, 64240, 33904800, 61721081184, 394586260943616, 9146766152111641344, 792073976107698469670400, 261895415169919230764987845120, 335402460348866803020064114666616832, 1678893205649791601327398844631544110815232
Offset: 1

Views

Author

Andrew Howroyd, Jan 16 2022

Keywords

Comments

This sequence differs from A049524 in that the source and sink are restricted to being single nodes.

Links

Crossrefs

The unlabeled version is A350794.
Row sums of A350791.
Cf. A049524, A165950, A350792, A003030.

Programs

  • Mathematica
    nn = 15; strong = Select[Import["https://oeis.org/A003030/b003030.txt", "Table"],
   Length@# == 2 &][[All, 2]]; s[z_] := Total[strong Table[z^i/i!, {i, 1, 58}]];
ggf[egf_] := Normal[Series[egf, {z, 0, nn}]] /. Table[z^i -> z^i/2^Binomial[i, 2], {i, 1, nn + 1}];egf[ggf_] := Normal[Series[ggf, {z, 0, nn}]] /.Table[z^i -> z^i*2^Binomial[i, 2], {i, 1, nn + 1}];Table[n!, {n, 0, nn}] CoefficientList[
Series[z - z^2 + Exp[(u - 1) (v - 1) s[ z]] egf[ggf[z Exp[(u - 1) s[z]]]*1/ggf[Exp[-s[z]]]*ggf[z Exp[(v - 1) s[ z]]]] /. {u -> 0, v -> 0}, {z, 0, nn}], z] (* Geoffrey Critzer, Apr 17 2023 *)
  • PARI
    InitFinallyV(12) \\ See A350791 for program code.

Formula

For n >= 3, a(n) = 2*n*(n-1)*A003030(n-1) (Robinson equation 22). - Geoffrey Critzer, Apr 17 2023

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))) }

A362013 Triangular array read by rows. T(n,k) is the number of labeled directed graphs on [n] with exactly k strongly connected components of size 1 with outdegree zero, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 27, 27, 9, 1, 2401, 1372, 294, 28, 1, 759375, 253125, 33750, 2250, 75, 1, 887503681, 171774906, 13852815, 595820, 14415, 186, 1, 3938980639167, 437664515463, 20841167403, 551353635, 8751645, 83349, 441, 1, 67675234241018881, 4263006881324024, 117484441611292, 1850148686792, 18210124870, 114709448, 451612, 1016, 1
Offset: 0

Views

Author

Geoffrey Critzer, Apr 03 2023

Keywords

Examples

			Triangle T(n,k) begins:
       1;
       0,      1;
       1,      2,     1;
      27,     27,     9,    1;
    2401,   1372,   294,   28,  1;
  759375, 253125, 33750, 2250, 75, 1;
  ...

Links

Crossrefs

Cf. A086206 (column k=0), A053763 (row sums), A361592, A350792 (a subclass of the digraphs for the case k=1 of this sequence), A003028.

Programs

  • Mathematica
    nn = 6; B[n_] := n! 2^Binomial[n, 2] ; strong =Select[Import["https://oeis.org/A003030/b003030.txt", "Table"], Length@# == 2 &][[All, 2]]; s[z_] := Total[strong Table[z^i/i!, {i, 1, 58}]];
ggf[egf_] := Normal[Series[egf, {z, 0, nn}]] /.Table[z^i -> z^i/2^Binomial[i, 2], {i, 0, nn}]; Table[ Take[(Table[B[n], {n, 0, nn}] CoefficientList[   Series[ggf[Exp[(u - 1) z]]/ggf[Exp[-s[z]]], {z, 0, nn}], {z, u}])[[i]], i], {i, 1, nn + 1}]
Showing 1-5 of 5 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.