A049531 -id:A049531 - cp's OEIS frontend

A350794 Number of digraphs on n unlabeled nodes with a global source and sink.

1, 1, 2, 20, 574, 48854, 12444800, 9849180230, 25265372689314, 218451490123178684, 6562780921564838071734, 700270642102506752862044142, 269621199012416753533007480951824, 378982029174285293421133982496722212766, 1962119228020498122395242424575089505014761082

Offset: 1

Andrew Howroyd, Jan 19 2022

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
Andrew Howroyd, PARI Program, Jan 2022, updated Mar 2023.

The labeled version is A350790.
Row sums of A350795.
Cf. A049531, A051421, A345258, A350360, A350796.

A350794seq(15) \\ See link for program code.

A051421 Number of digraphs on n unlabeled nodes with a sink (or, with a source).

Original entry on oeis.org

1, 2, 12, 185, 8990, 1505939, 875542491, 1789247738400, 13018820342147705, 341188114831706152794, 32520852428719860881939391, 11366533535523591133597276823755, 14669006027884671703581740693080811331, 70315546525961698601351615055416574931833334

Offset: 1

Vladeta Jovovic

Here a sink is defined to be a node to which all other nodes have a directed path. - Andrew Howroyd, Dec 27 2021

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 218 (incorrect version).
Ronald C. Read, email to N. J. A. Sloane, 28 August, 2000.

Andrew Howroyd, Table of n, a(n) for n = 1..50

The labeled case is A003028.
Row sums of A057277.
Cf. A003085, A035512, A003088, A049531, A350415, A350360, A350794.

\\ See PARI link in A350794 for program code.
A051421seq(15) \\ Andrew Howroyd, Jan 21 2022

a(6) corrected and a(7) from Sean A. Irvine, Sep 11 2021

a(0)=1 removed and terms a(8) and beyond from Andrew Howroyd, Jan 21 2022

A057278 Triangle T(n,k) of number of digraphs with a source and a sink on n unlabeled nodes and k arcs, k=0..n*(n-1).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 4, 4, 1, 1, 0, 0, 0, 1, 11, 31, 45, 38, 27, 13, 5, 1, 1, 0, 0, 0, 0, 1, 23, 152, 486, 992, 1419, 1641, 1485, 1152, 707, 379, 154, 61, 16, 5, 1, 1, 0, 0, 0, 0, 0, 1, 42, 517, 3194, 12174, 32860, 68423, 116168, 166164, 204867, 219906, 206993, 170922, 124088, 78809, 43860, 21209, 8951, 3242, 1043, 288, 76, 17, 5, 1, 1

Offset: 1

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

			Triangle begins:
  [1],
  [0,1,1],
  [0,0,1,4,4,1,1],
  [0,0,0,1,11,31,45,38,27,13,5,1,1],
  ...
The number of digraphs with a source and a sink on 3 unlabeled nodes is 11 = 1+4+4+1+1.

V. Jovovic, G. Kilibarda, Enumeration of labeled initially-finally connected digraphs, Scientific review, Serbian Scientific Society, 19-20 (1996), p. 246.

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)

Row sums give A049531.
Column sums give A350906.
The labeled version is A057271.
Cf. A057270, A057276, A057277, A057279, A350794, A350795.

\\ See PARI link in A350794 for program code.
{ my(A=A057278triang(5)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 21 2022

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

A345258 Number of acyclic digraphs (or DAGs) on n unlabeled vertices with one source and one sink.

Original entry on oeis.org

1, 1, 2, 10, 98, 1960, 80176, 6686760, 1129588960, 384610774696, 263104175114712, 360908867732030980, 991603865814038728388, 5453395569997436383751204, 60010050181461052836515513108, 1321051495313052133670927704328040, 58170762510305449187073353930875222256

Offset: 1

Max Alekseyev, Jun 12 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50 (terms 1..40 from Mikhail Tikhomirov)
Marcel et al., Is there a formula for the number of st-dags (DAG with 1 source and 1 sink) with n vertices?, MathOverflow, 2021.

Row sums of A350491.
The labeled version is A165950.
Cf. A003087, A049531, A122078, A350415, A350492.

A345258seq(16) \\ See PARI link in A122078 for program code.

a(9) from Brendan McKay.

Terms a(10) and beyond from Mikhail Tikhomirov, Jun 16 2021

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

Jim Snyder-Grant, Dec 26 2021

nonn

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.

			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}

Andrew Howroyd, Table of n, a(n) for n = 1..50
Jim Snyder-Grant, C code to generate and count digraphs with global sinks
Eric Weisstein's World of Mathematics, Digraph Sink

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

\\ See PARI link in A350794 for program code.
A350360seq(15) \\ Andrew Howroyd, Jan 21 2022

# 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)]

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

A049524 Number of digraphs with a source and a sink on n labeled nodes.

Original entry on oeis.org

1, 3, 48, 3424, 962020, 1037312116, 4344821892264, 71771421308713624, 4716467927380427847264, 1237465168798883061207535456, 1297923989772809185944542332007104, 5444330658513426322624322033259452670016, 91342931436147421630261703458729460990513248512

Offset: 1

Vladeta Jovovic, Goran Kilibarda

nonn

Here a source is defined to be a node which has a directed path to all other nodes and a sink to be a node to which all other nodes have a directed path. A digraph with a source and a sink can also be described as initially-finally connected. - Andrew Howroyd, Jan 16 2022

V. Jovovic, G. Kilibarda, Enumeration of labeled initially-finally connected digraphs, Scientific review, Serbian Scientific Society, 19-20 (1996), p. 244.

Andrew Howroyd, Table of n, a(n) for n = 1..50
Sean A. Irvine, Java program (github)
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.

The unlabeled version is A049531.
Row sums of A057271.
Cf. A003027, A003028, A003029, A003030, A049414, A350790.

InitFinally(15) \\ See A057271. - Andrew Howroyd, Jan 16 2022

Terms a(12) and beyond from Andrew Howroyd, Jan 16 2022

A003088 Number of unilateral digraphs with n unlabeled nodes.

Original entry on oeis.org

1, 1, 2, 11, 171, 8603, 1478644, 870014637

Offset: 0

N. J. A. Sloane

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 218.
Ronald C. Read, email to N. J. A. Sloane, 28 August, 2000.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. W. Robinson, Counting strong digraphs (research announcement), J. Graph Theory 1, 1977, pp. 189-190.

Cf. A057270, A003029 (labeled case), A003085, A035512, A051421, A049531, A049512.

Note that Read and Wilson incorrectly give a(4) as 172 - thanks to Vladeta Jovovic, Goran Kilibarda for finding this error and for verifying a(5).

a(7) from Sean A. Irvine, Jan 26 2015

A350906 Number of unlabeled initially-finally connected digraphs with n arcs.

Original entry on oeis.org

1, 1, 2, 5, 16, 56, 241, 1111, 5748, 31912, 190328, 1204317, 8050341, 56516303, 415128256, 3179020729, 25308005049, 208918893640, 1784451646305, 15739926014382, 143130061546156, 1339761466105929, 12891329514544223, 127351374611721837, 1290198076415425548, 13390893744153374776

Offset: 0

Andrew Howroyd, Jan 21 2022

nonn

Column sums of A057278.

Cf. A049531 (by vertices), A350752, A350796, A350798, A350907.

A350906seq(15) \\ See PARI link in A350794 for program code.

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A350794 Number of digraphs on n unlabeled nodes with a global source and sink.

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A051421 Number of digraphs on n unlabeled nodes with a sink (or, with a source).

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A057278 Triangle T(n,k) of number of digraphs with a source and a sink on n unlabeled nodes and k arcs, k=0..n*(n-1).

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A345258 Number of acyclic digraphs (or DAGs) on n unlabeled vertices with one source and one sink.

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

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A049524 Number of digraphs with a source and a sink on n labeled nodes.

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A003088 Number of unilateral digraphs with n unlabeled nodes.

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A350906 Number of unlabeled initially-finally connected digraphs with n arcs.

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