A135388 Number of (directed) Eulerian circuits on the complete graph K_{2n+1}.

2, 264, 129976320, 911520057021235200, 257326999238092967427785160130560, 6705710151431658873046319662156165939200000000000000, 32132958735643556926111996291480203406145819659840760945049600000000000000000

Offset: 1

Author

Max Alekseyev, Dec 10 2007

Keywords

• nonn
• walk

References

• B. D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221.

Links

• Max Alekseyev, Table of n, a(n) for n = 1..10
• Brendan D. McKay and Robert W. Robinson, Asymptotic enumeration of Eulerian circuits in the complete graph, Combinatorics, Probability and Computing, 7 (1998), 437-449.
• G. Tarry, Géométrie de situation: Nombre de manières distinctes de parcourir en une seule course toutes les allées d'un labyrinthe rentrant, en ne passant qu'une seule fois par chacune des allées, Comptes Rendus Assoc. Franç. Avance. Sci. 15, part 2 (1886), pp. 49-53.
• Eric Weisstein's World of Mathematics, Complete Graph
• Eric Weisstein's World of Mathematics, Eulerian Cycle

Crossrefs

Bisection of A350028.

Cf. A007082, A232545, A357856, A357857, A356366, A357855, A357885, A357886, A357887.

Cf. A369820 (undirected Eulerian circuits).

Programs

Mathematica

Table[2 Length[FindEulerianCycle[CompleteGraph[2 n + 1], All]], {n, 3}] (* Eric W. Weisstein, Jan 09 2018 *)
  (* a(3) requires a very large amount of memory *)

Formula

a(n) = A007082(n) * (n-1)!^(2*n+1).

a(n) = A350028(2n+1) = A357887(2n+1,n(2n+1)). - Max Alekseyev, Oct 19 2022