A382212 Number of labeled Eulerian oriented graphs with n nodes without isolated vertices.

1, 0, 0, 2, 6, 168, 6700, 726360, 202827786, 154606199048, 326652738166674, 1943277939095470560, 32810852170233531563092, 1582383093047152724860680720, 218967451207911981699586124870064, 87266156428964732861556013539795735380

Offset

0,4

Comments

This sequence is similar to A007081 but imposes the restriction that the graphs contain no isolated vertices.

Each graph can be considered to represent a generalized rock-paper-scissors game for which a uniformly random strategy is optimal and no draw can be forced.

Links

Table of n, a(n) for n=0..15.

Wikipedia, Rock paper scissors

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A007081(k). - Max Alekseyev, Oct 25 2025

Example

For n=3, the only solutions are given by the directed cycles (1,2,3) and (1,3,2). Similarly, also n=4 only has directed cycles as solutions.
An example of a solution for n=5 is
  1 -> {5}
  2 -> {5}
  3 -> {4}
  4 -> {1,2}
  5 -> {3,4}
Disconnected solutions exist only for n >= 6.

Crossrefs

Cf. A007081.

Sequence in context: A281819 A329980 A199482 * A168649 A135937 A182523

Adjacent sequences: A382209 A382210 A382211 * A382213 A382214 A382215

Keyword

nonn

Author

Bert Dobbelaere, Mar 18 2025

Extensions

a(0) prepended and a(9)-a(15) added by Max Alekseyev, Oct 25 2025

Status

approved