An improper tournament, or tournament with ties, is a graph in which every pair of nodes is connected by a single uniquely directed edge or by a single undirected edge.
There are 1, 2, and 7 improper tournaments of orders 1, 2, and 3, respectively. How many are there of order 4? Of order n?



Counting tournaments with ties
What you are looking for in the number of [oriented graphs](http://mathworld.wolfram.com/OrientedGraph.html) on $n$ vertices. Just think of an undirected edge in an improper tournament as a missing edge in an oriented graph and vice versa. The number of oriented graphs on $n$ vertices is [OEIS A001174](https://oeis.org/A001174). Note there are $7$ oriented graphs for $n=3$ vertices (not $6$). All $7$ are shown in the first link.


You may also be interested in the paper [The number of oriented graphs](http://projecteuclid.org/euclid.mmj/1028997952) by Frank Harary. There are also the papers [Asymptotic formulas for the number of oriented graphs](http://www.sciencedirect.com/science/article/pii/S009589567680010X) and [Asymptotic formulas for the number of oriented graphs](http://www.sciencedirect.com/science/article/pii/0095895678900588) (two papers both with same title).


