A308111 -id:A308111 - cp's OEIS frontend

A229865 Number of n X n 0..1 arrays with corresponding row and column sums equal.

1, 2, 8, 80, 2432, 247552, 88060928, 112371410944, 523858015518720, 9041009511609073664, 583447777113052431515648, 141885584718620229407228821504, 130832005909904417592540055577034752, 459749137931232137234615429529864283095040, 6182706200522446492946534924719926752508110700544

Offset: 0

R. H. Hardin, Oct 01 2013

nonn

Also known as labeled Eulerian digraphs allowing loops. - Brendan McKay, May 12 2019

			Some solutions for n=4:
  0 0 0 1     0 0 1 0     0 0 0 1     0 0 1 0     0 0 1 1
  0 1 0 0     1 0 0 0     1 0 1 0     0 0 1 1     1 0 0 1
  0 0 0 1     0 1 0 0     0 1 0 1     0 1 1 1     1 1 1 0
  1 0 1 0     0 0 0 1     0 1 1 0     1 1 0 0     0 1 1 1
From _Gus Wiseman_, Jun 22 2019: (Start)
The a(3) = 8 Eulerian digraph edge-sets:
  {}
  {11}
  {22}
  {11,22}
  {12,21}
  {11,12,21}
  {12,21,22}
  {11,12,21,22}
(End)

Mohammad Behzad Kang and Andrew Salch, The mod p cohomology of the Morava stabilizer group at large primes, arXiv:2410.24171 [math.AT], 2024. See p. 46.

Column 1 of A229870.
The unlabeled version is A308111.
Cf. A000595, A002416, A002720, A007080.
Cf. A326209, A326237, A326251, A326252, A326253.

Table[Length[Select[Subsets[Tuples[Range[n],2]],Sort[First/@#]==Sort[Last/@#]&]],{n,4}] (* Gus Wiseman, Jun 22 2019 *)

a(n) = 2^n * A007080(n). - Andrew Howroyd, Sep 11 2019

a(0)=1 prepended by Alois P. Heinz, May 14 2019

Terms a(11) and beyond from Andrew Howroyd, Sep 11 2019

A058338 Number of digraphs with indegree = outdegree at each vertex, or Eulerian digraphs (including disconnected graphs) with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 18, 111, 2292, 181519, 51294836, 48814342597, 157166848652408, 1736132851915020181, 66689407510956897981608, 9012860249531358132125181323, 4328664025050045497712238253179872, 7451990930474005836163802713084971814275, 46329549163592383403451764167315165130121820112

Offset: 0

N. J. A. Sloane

Every regular tournament (A096368) is a Eulerian digraph. Similar methods may be used to compute terms of this sequence. - Andrew Howroyd, Apr 12 2020

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 219 (but there is an error).
Ronald C. Read, email to N. J. A. Sloane, 28 August, 2000.

Cf. A007080 (labeled), A058337 (connected), A096368, A308161, A308111.

A058337 = Cases[Import["https://oeis.org/A058337/b058337.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A058337] (* Jean-François Alcover, Feb 23 2020, upated Mar 18 2020 *)

Euler transform of A058337.

a(7) added using A058337 by Falk Hüffner, Dec 03 2015
a(8) and a(9) added using A058337 by Brendan McKay, May 05 2019
Terms a(10) and beyond from Andrew Howroyd, Apr 12 2020

A308161 Number of isomorphism classes of Eulerian oriented graphs with n vertices.

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 24, 200, 5479, 439517, 91097868, 48916220147, 68628518786683, 254305521019154638, 2512451288680194070842, 66741359152815902974086530, 4802230893555589082929258033462, 942013815025325986980154281918094498, 506666364226468633163453153303288094604018

Offset: 0

Brendan McKay, May 15 2019

Loops and 2-cycles are not permitted. Eulerian means that each vertex has equal in-degree and out-degree.

			For n=4, the a(4)=3 solutions are an empty graph, a directed 3-cycle plus an isolated vertex, and a directed 4-cycle.

A058338 is the same allowing 2-cycles.
A308111 is the same allowing both loops and 2-cycles.
Cf. A007081 (labeled), A308239 (connected).

Terms a(12) and beyond from Andrew Howroyd, Apr 10 2020

A308128 Isomorphism classes of connected Eulerian digraphs with n vertices, allowing loops.

Original entry on oeis.org

1, 2, 3, 14, 112, 2174, 124501, 22400498, 13010949171, 24926846389076, 160810397320789069, 3554744065897655673978, 273140190737719436311559660, 73831786956788218320014098284918, 70920392983384812245087697080226658475, 244186402549448674084991238687021028510453186

Offset: 0

Alois P. Heinz, May 14 2019, following a suggestion from Brendan McKay

Cf. A058338, A229865, A308111.

Inverse Euler transform of A308111.

Terms a(10) and beyond from Andrew Howroyd, Apr 12 2020

cp's OEIS Frontend

A229865 Number of n X n 0..1 arrays with corresponding row and column sums equal.

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A058338 Number of digraphs with indegree = outdegree at each vertex, or Eulerian digraphs (including disconnected graphs) with n nodes.

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A308161 Number of isomorphism classes of Eulerian oriented graphs with n vertices.

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A308128 Isomorphism classes of connected Eulerian digraphs with n vertices, allowing loops.

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