A058338 -id:A058338 - cp's OEIS frontend

A007080 Number of labeled Eulerian digraphs with n nodes.

1, 2, 10, 152, 7736, 1375952, 877901648, 2046320373120, 17658221702361472, 569773219836965265152, 69280070663388783890248448, 31941407692847758201303724506112, 56121720938871110502272391300032261120, 377362438996731353329256282026362716827887616, 9744754031799754169218003376206941877943086188308480, 969342741943194323476512925742876053501022995325734477987840

Offset: 1

N. J. A. Sloane, Mira Bernstein

Includes disconnected graphs. - Felix A. Pahl, Jul 15 2018

Loops and parallel edges are not allowed, but 2-cycles (in other words, edges A --> B and B --> A existing simultaneously) are allowed. - Mikhail Lavrov, Mar 04 2025

			For n=3, the a(n) = 10 solutions are: (A . B . C), (A <--> B . C), (A <--> C . B), (B <--> C . A), (A --> B --> C --> A), (A --> C --> B --> A), (A <--> B <--> C), (A <--> C <--> B), (B <--> A <--> C), and (A <--> B <--> C <--> A). - _Mikhail Lavrov_, Mar 04 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Matteo Beccaria, Thermal properties of a string bit model at large N, arXiv:1709.01801 [hep-th], 2017.
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.
Rüdiger Jehn, Kester Habermann, and Misha Lavrov, Number of ways that a football league can complete with all teams having the same number of points, arXiv:2503.14509 [math.GM], 2025.
B. D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221.
B. D. McKay, The asymptotic numbers of regular tournaments, Eulerian digraphs and Eulerian oriented graphs, Combinatorica 10 (1990), 367-377.

Cf. A058338 (unlabeled), A229865 (loops allowed), A054957 (connected) - Mikhail Lavrov, Mar 04 2025

a[n_]:=Coefficient[Expand[Product[Product[x[i]+x[j],{j, 1, n}],{i, 1,n}]],Product[x[k]^n,{k,1,n}]]/2^n (* practically unusable for n>7 *)
a[n_]:=N[(Sqrt[n]/E^(1/4))*(2^n/Sqrt[n*Pi])^(n-1)*(1+3/(16*n)+1/(7*n^2)+3/(20*n^3))]
(* four digit accuracy for n>7 *) (* Thomas Curtright, Apr 12 2017 *)

a(n) ~ e^(-1/4)*sqrt(n)(2^n/sqrt(Pi*n))^(n-1)*(1+O(1/sqrt(n))) [B. D. McKay, 1990]. - Thomas Curtright, Apr 11 2017

Terms a(12) and beyond from McKay (1983), added by Thomas Curtright, Apr 12 2017

A058337 Number of connected loop-free Eulerian digraphs with n nodes.

Original entry on oeis.org

1, 1, 1, 3, 12, 90, 2162, 179098, 51110788, 48762860958, 157117982629089, 1735975636147643118, 66687671220840028265672, 9012793558387399546070260538, 4328655012123103085159161399497333, 7451986601800967792474321528025564017244, 46329541711597124247394557123740768918978380463

Offset: 0

N. J. A. Sloane

Parallel edges are not allowed, but 2-cycles are allowed. Since these digraphs have an Eulerian circuit, being weakly-connected is the same as being strongly-connected. - Brendan McKay, Apr 04 2023

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.

Eric Weisstein's World of Mathematics, Eulerian Graph

Cf. A058338.

Inverse Euler transform of A058338. - Andrew Howroyd, Apr 12 2020

a(7) (generated graphs with nauty, filtered and counted with Python) by Pietro Battiston, Feb 08 2014
a(8) and a(9) added and description clarified by Brendan McKay, May 05 2019
Terms a(10) and beyond from Andrew Howroyd, Apr 12 2020

A308111 Isomorphism classes of Eulerian digraphs with n vertices, allowing loops.

Original entry on oeis.org

1, 2, 6, 24, 160, 2512, 129816, 22665792, 13056562208, 24953006054144, 160860329639968800, 3555065836569542246400, 273147301191314006316868352, 73832333258502021627712839197696, 70920540648597652305602460997787710080, 244186544390677638132290202415190606165938176, 3036252267734950687777830287721323374283100639476736

Offset: 0

Brendan McKay, May 11 2019

Eulerian means that for every vertex the in-degree equals the out-degree.

			For n=2 the a(2)=6 solutions are: two non-adjacent vertices with or without loops (3 cases), two vertices with or without loops connected by edges in each direction (3 cases).

Gus Wiseman, Non-isomorphic representatives of the a(3) = 24 Eulerian digraphs with loops.

For labeled digraphs rather than isomorphism classes see A229865.
For isomorphism classes with loops forbidden see A058338.
Cf. A308128 (connected version of this).
Cf. A000273, A000595, A002416, A002720, A308161.

Euler transform of A308128.

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

A054957 Number of labeled connected Eulerian digraphs with n nodes.

Original entry on oeis.org

1, 1, 6, 118, 7000, 1329496, 868255024, 2039295163312, 17639804273910144, 569596637165777524096, 69273803156588266525129984, 31940576331984155835757999795456, 56121305700570450429977355284230355968, 377361653292638108206937443603065676106341376

Offset: 1

N. J. A. Sloane, May 24 2000

Alois P. Heinz, Table of n, a(n) for n = 1..16
V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A058338, A054955, A007080 (EXP-transform), A054959, A054956, A058337, A054958.

a(n) = A054958(n) + A054959(n). - R. J. Mathar, Apr 13 2017

More terms from Vladeta Jovovic, Mar 06 2001

More terms from Alois P. Heinz, Apr 14 2017

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

A054958 Number of labeled connected Eulerian digraphs with n nodes and an even number of edges.

Original entry on oeis.org

1, 1, 4, 62, 3512, 664808, 434127872, 1019647584176, 8819902136975232, 284798318582888943488, 34636901578294133264379392, 15970288165992077917879019856128, 28060652850285225214988677642354678784, 188680826646319054103468721801532841166681088

Offset: 1

N. J. A. Sloane, May 24 2000

V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A058338, A054955, A007080, A054959, A054956, A058337, A054957.

A054957 = Cases[Import["https://oeis.org/A054957/b054957.txt", "Table"], {, }][[All, 2]];
a[n_] := (A054957[[n]] + (n - 1)!)/2;
Array[a, 14] (* Jean-François Alcover, Sep 13 2019 *)

a(n) = (A054957 + (n-1)!)/2. - Andrew Howroyd, Sep 10 2018

More terms from Vladeta Jovovic, Mar 06 2001

a(12)-a(14) from Andrew Howroyd, Sep 10 2018

A054959 Number of labeled connected Eulerian digraphs with n nodes and an odd number of edges.

Original entry on oeis.org

0, 0, 2, 56, 3488, 664688, 434127152, 1019647579136, 8819902136934912, 284798318582888580608, 34636901578294133260750592, 15970288165992077917878979939328, 28060652850285225214988677641875677184, 188680826646319054103468721801532834939660288

Offset: 1

N. J. A. Sloane, May 24 2000

V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A058338, A054955, A007080, A054956, A058337, A054958, A054957.

A054957 = Cases[Import["https://oeis.org/A054957/b054957.txt", "Table"], {, }][[All, 2]];
a[n_] := (A054957[[n]] - (n - 1)!)/2;
Array[a, 14] (* Jean-François Alcover, Sep 13 2019 *)

a(n) = (A054957(n) - (n-1)!)/2. - Andrew Howroyd, Sep 10 2018

More terms from Vladeta Jovovic, Mar 06 2001

a(12)-a(14) from Andrew Howroyd, Sep 10 2018

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

A007080 Number of labeled Eulerian digraphs with n nodes.

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A058337 Number of connected loop-free Eulerian digraphs with n nodes.

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A308111 Isomorphism classes of Eulerian digraphs with n vertices, allowing loops.

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A054957 Number of labeled connected Eulerian digraphs with n nodes.

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

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A054958 Number of labeled connected Eulerian digraphs with n nodes and an even number of edges.

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A054959 Number of labeled connected Eulerian digraphs with n nodes and an odd number of edges.

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

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