A380592 Number of ways that a European soccer league tournament with n teams can complete with all teams having the same number of points.

1, 3, 27, 1083, 296081, 696779523, 16503494334993, 3439079361325736243

Offset: 1

Author

Ruediger Jehn, Jan 27 2025

Keywords

• nonn
• hard
• more

Comments

Teams play each other twice for a total of M = n*(n-1) matches.

A victory is awarded 3 points, a draw 1 point and a defeat 0 points.

The total number of possible match outcomes is 3^M = A053764(n) and a(n) is how many of them result in all teams finishing with the same points score.

If all matches were randomly assigned a result, the probability that all teams would end up with the same number of points is a(n)/A053764(n), which in a typical league of 18 or 20 teams is very small.

A007080(n) is the number of ways if there are no draws.

Examples

			We denote the vector (r1, r2 ... r_M) with r_i in {0, 1, 3} as a possible sequence of match results. Then a(2) = 3: (0, 0) - both teams lose their home game and have 3 points at the end, (1,1) - both matches end with a draw and both teams have 2 points, (3,3) - both teams win their home game and have 3 points.
		

Links

• Ruediger Jehn, Julia brute-force program
• 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.
• Ruediger Jehn, Julia program based on fast algorithm

Crossrefs

Cf. A053764, A007080.