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

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• nice

Comments

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

Examples

			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
		

References

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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

Extensions

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