This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380592 #35 Mar 25 2025 19:52:02
%S A380592 1,3,27,1083,296081,696779523,16503494334993,3439079361325736243
%N A380592 Number of ways that a European soccer league tournament with n teams can complete with all teams having the same number of points.
%C A380592 Teams play each other twice for a total of M = n*(n-1) matches.
%C A380592 A victory is awarded 3 points, a draw 1 point and a defeat 0 points.
%C A380592 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.
%C A380592 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.
%C A380592 A007080(n) is the number of ways if there are no draws.
%H A380592 Ruediger Jehn, <a href="/A380592/a380592.jl.txt">Julia brute-force program</a>
%H A380592 Rüdiger Jehn, Kester Habermann, and Misha Lavrov, <a href="https://arxiv.org/abs/2503.14509">Number of ways that a football league can complete with all teams having the same number of points</a>, arXiv:2503.14509 [math.GM], 2025.
%H A380592 Ruediger Jehn, <a href="/A380592/a380592_2.jl.txt">Julia program based on fast algorithm</a>
%e A380592 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.
%Y A380592 Cf. A053764, A007080.
%K A380592 nonn,hard,more
%O A380592 1,2
%A A380592 _Ruediger Jehn_, Jan 27 2025