A350415 -id:A350415 - cp's OEIS frontend

A003025 Number of n-node labeled acyclic digraphs with 1 out-point.

1, 2, 15, 316, 16885, 2174586, 654313415, 450179768312, 696979588034313, 2398044825254021110, 18151895792052235541515, 299782788128536523836784628, 10727139906233315197412684689421

Offset: 1

N. J. A. Sloane

nonn

From Gus Wiseman, Jan 02 2024: (Start)
Also the number of n-element sets of finite nonempty subsets of {1..n}, including a unique singleton, such that there is exactly one way to choose a different element from each. For example, the a(0) = 0 through a(3) = 15 set-systems are:
  .  {{1}}  {{1},{1,2}}  {{1},{1,2},{1,3}}
            {{2},{1,2}}  {{1},{1,2},{2,3}}
                         {{1},{1,3},{2,3}}
                         {{2},{1,2},{1,3}}
                         {{2},{1,2},{2,3}}
                         {{2},{1,3},{2,3}}
                         {{3},{1,2},{1,3}}
                         {{3},{1,2},{2,3}}
                         {{3},{1,3},{2,3}}
                         {{1},{1,2},{1,2,3}}
                         {{1},{1,3},{1,2,3}}
                         {{2},{1,2},{1,2,3}}
                         {{2},{2,3},{1,2,3}}
                         {{3},{1,3},{1,2,3}}
                         {{3},{2,3},{1,2,3}}
These set-systems are all connected.
The case of labeled graphs is A000169.
(End)

			a(2) = 2: o-->--o (2 ways)
a(3) = 15: o-->--o-->--o (6 ways) and
o ... o o-->--o
.\ . / . \ . /
. v v ... v v
.. o ..... o
(3 ways) (6 ways)

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).

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

A diagonal of A058876.
Row sums of A350487.
The unlabeled version is A350415.
Column k=1 of A361718.
Cf. A003026, A003028, A345258.
For any number of sinks we have A003024, unlabeled A003087.
For n-1 sinks we have A058877.
For a fixed sink we have A134531 (up to sign), column k=1 of A368602.
Cf. A000169, A058891, A059201, A082402, A088957, A323818, A334282, A367904, A368600, A368601.

Table[Length[Select[Subsets[Subsets[Range[n]],{n}],Count[#,{}]==1&&Length[Select[Tuples[#],UnsameQ@@#&]]==1&]],{n,0,4}] (* _Gus Wiseman, Jan 02 2024 *)

\\ requires A058876.
my(T=A058876(20)); vector(#T, n, T[n][1]) \\ Andrew Howroyd, Dec 27 2021

\\ see Marcel et al. link (formula for a').
seq(n)={my(a=vector(n)); a[1]=1; for(n=2, #a, a[n]=sum(k=1, n-1, (-1)^(k-1) * binomial(n,k) * (2^(n-k)-1)^k * a[n-k])); a} \\ Andrew Howroyd, Jan 01 2022

a(n) = (-1)^(n-1) * n * A134531(n). - Gus Wiseman, Jan 02 2024

More terms from Vladeta Jovovic, Apr 10 2001

A122078 Triangle read by rows: T(n,k) is the number of unlabeled acyclic digraphs with n >= 0 nodes and n-k outnodes (0 <= k <= n).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 11, 16, 0, 1, 4, 25, 108, 164, 0, 1, 5, 47, 422, 2168, 3341, 0, 1, 6, 78, 1251, 15484, 88747, 138101, 0, 1, 7, 120, 3124, 79836, 1215783, 7409117, 11578037, 0, 1, 8, 174, 6925, 333004, 11620961, 199203464, 1252610909, 1961162564, 0

Offset: 0

N. J. A. Sloane, Oct 18 2006

			Triangle T(n,k) begins:
  1:
  1, 0;
  1, 1,  0;
  1, 2,  3,    0;
  1, 3, 11,   16,     0;
  1, 4, 25,  108,   164,     0;
  1, 5, 47,  422,  2168,  3341,      0;
  1, 6, 78, 1251, 15484, 88747, 138101, 0;
  ...

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

Andrew Howroyd, Table of n, a(n) for n = 0..495 (rows 0..30; rows 0..15 from R. W. Robinson)
Andrew Howroyd, PARI program, Dec 2021, updated Jan 2022.

Row sums give A003087.
Diagonals include A000007, A350415.
Cf. A058876 (labeled case), A350447, A350448, A350449, A350450.

\\ See link for program code.
{ my(T=AcyclicDigraphsByNonSources(8)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Dec 31 2021

Zero terms inserted by Andrew Howroyd, Dec 29 2021

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

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

A350488 Triangle read by rows: T(n,k) is the number of acyclic digraphs on n unlabeled nodes with k arcs and a global source, n >= 1, k = 0..n*(n-1)/2.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 4, 6, 5, 1, 0, 0, 0, 0, 9, 25, 47, 46, 27, 9, 1, 0, 0, 0, 0, 0, 20, 95, 297, 582, 783, 738, 501, 235, 75, 14, 1, 0, 0, 0, 0, 0, 0, 48, 337, 1575, 4941, 11295, 19404, 25847, 26966, 22195, 14380, 7280, 2831, 816, 165, 20, 1

Offset: 1

Andrew Howroyd, Jan 01 2022

			Triangle begins:
  [1] 1;
  [2] 0, 1;
  [3] 0, 0, 2, 1;
  [4] 0, 0, 0, 4, 6,  5,  1;
  [5] 0, 0, 0, 0, 9, 25, 47, 46, 27, 9, 1;
  [6] 0, 0, 0, 0, 0, 20, 95, 297, 582, 783, 738, 501, 235, 75, 14, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1350 (rows 1..20)

Row sums are A350415.
Column sums are A350490.
Leading diagonal is A000081.
The labeled version is A350487.
Cf. A122078, A350447, A350449, A350491.

\\ See PARI link in A122078 for program code.
{ my(A=A350488rows(7)); for(i=1, #A, print(A[i])) }

A350490 Number of unlabeled acyclic digraphs with n arcs and a global source.

Original entry on oeis.org

1, 1, 2, 5, 15, 50, 191, 795, 3625, 17781, 93319, 519732, 3056185, 18880345, 122066437, 823118881, 5772473719, 41995743085, 316258151517, 2460542117337, 19743544526463, 163138014893817, 1386173917051704, 12096681529032989, 108293699332498981

Offset: 0

Andrew Howroyd, Jan 08 2022

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column sums of A350488.

Cf. A122078, A350415, A350451, A350492.

A350490seq(20) \\ See PARI link in A122078 for program code.

A361718 Triangular array read by rows. T(n,k) is the number of labeled directed acyclic graphs on [n] with exactly k nodes of indegree 0.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 15, 9, 1, 0, 316, 198, 28, 1, 0, 16885, 10710, 1610, 75, 1, 0, 2174586, 1384335, 211820, 10575, 186, 1, 0, 654313415, 416990763, 64144675, 3268125, 61845, 441, 1, 0, 450179768312, 286992935964, 44218682312, 2266772550, 43832264, 336924, 1016, 1

Offset: 0

Geoffrey Critzer, Apr 02 2023

Also the number of sets of n nonempty subsets of {1..n}, k of which are singletons, such that there is only one way to choose a different element from each. For example, row n = 3 counts the following set-systems:
  {{1},{1,2},{1,3}}    {{1},{2},{1,3}}    {{1},{2},{3}}
  {{1},{1,2},{2,3}}    {{1},{2},{2,3}}
  {{1},{1,3},{2,3}}    {{1},{3},{1,2}}
  {{2},{1,2},{1,3}}    {{1},{3},{2,3}}
  {{2},{1,2},{2,3}}    {{2},{3},{1,2}}
  {{2},{1,3},{2,3}}    {{2},{3},{1,3}}
  {{3},{1,2},{1,3}}    {{1},{2},{1,2,3}}
  {{3},{1,2},{2,3}}    {{1},{3},{1,2,3}}
  {{3},{1,3},{2,3}}    {{2},{3},{1,2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{1,3},{1,2,3}}
  {{2},{1,2},{1,2,3}}
  {{2},{2,3},{1,2,3}}
  {{3},{1,3},{1,2,3}}
  {{3},{2,3},{1,2,3}}

			Triangle begins:
  1;
  0,     1;
  0,     2,     1;
  0,    15,     9,    1;
  0,   316,   198,   28,  1;
  0, 16885, 10710, 1610, 75, 1;
  ...

E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.
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.

Cf. A058876 (mirror), A361579, A224069.
Row-sums are A003024, unlabeled A003087.
Column k = 1 is A003025(n) = |n*A134531(n)|.
Column k = n-1 is A058877.
For fixed sinks we get A368602.
A058891 counts set-systems, unlabeled A000612.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A000169, A059201, A082402, A088957, A133686, A334282, A350415, A367904, A367908, A368600, A368601.

nn = 8; B[n_] := n! 2^Binomial[n, 2] ;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[-z]], {z, 0, nn}], {z, u}])[[i]], i], {i, 1, nn + 1}] // Grid
nv=4;Table[Length[Select[Subsets[Subsets[Range[n]],{n}], Count[#,{_}]==k&&Length[Select[Tuples[#], UnsameQ@@#&]]==1&]],{n,0,nv},{k,0,n}]

T(n,k) = A368602(n,k) * binomial(n,k). - Gus Wiseman, Jan 03 2024

A368602 Triangle read by rows where T(n,k) is the number of labeled acyclic digraphs on {1..n} with sinks {1..k}.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 3, 1, 0, 79, 33, 7, 1, 0, 3377, 1071, 161, 15, 1, 0, 362431, 92289, 10591, 705, 31, 1, 0, 93473345, 19856703, 1832705, 93375, 2945, 63, 1, 0, 56272471039, 10249747713, 789619327, 32382465, 782719, 12033, 127, 1

Offset: 0

Gus Wiseman, Jan 02 2024

Also the number of set-systems with n vertices and n edges such that {i} is a singleton edge iff i <= k, and such that there is only one way to choose a different vertex from each edge.

			Triangle begins:
    1
    0    1
    0    1    1
    0    5    3    1
    0   79   33    7    1
    0 3377 1071  161   15    1
    ...
Row n = 3 counts the following set-systems:
  {{1},{1,2},{1,3}}    {{1},{2},{1,3}}    {{1},{2},{3}}
  {{1},{1,2},{2,3}}    {{1},{2},{2,3}}
  {{1},{1,3},{2,3}}    {{1},{2},{1,2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{1,3},{1,2,3}}

Column k = n-1 is A000225 = A058877(n)/n.
Column k = 1 is A134531 (up to sign) or A003025(n)/n, non-fixed A350415.
For any choice of k sinks we get A361718.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A000169, A003024, A003087, A082402, A088957, A334282, A367862, A367904, A367908, A368600, A368601.

Table[Length[Select[Subsets[Subsets[Range[n]],{n}], Union@@Cases[#,{_}]==Range[k] && Length[Select[Tuples[#],UnsameQ@@#&]]==1&]], {n,0,3},{k,0,n}]

T(n,k) = A361718(n,k)/binomial(n,k).

More terms from Alois P. Heinz, Jan 04 2024

cp's OEIS Frontend

A003025 Number of n-node labeled acyclic digraphs with 1 out-point.

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A122078 Triangle read by rows: T(n,k) is the number of unlabeled acyclic digraphs with n >= 0 nodes and n-k outnodes (0 <= k <= n).

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

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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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A350488 Triangle read by rows: T(n,k) is the number of acyclic digraphs on n unlabeled nodes with k arcs and a global source, n >= 1, k = 0..n*(n-1)/2.

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A350490 Number of unlabeled acyclic digraphs with n arcs and a global source.

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A361718 Triangular array read by rows. T(n,k) is the number of labeled directed acyclic graphs on [n] with exactly k nodes of indegree 0.

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