A053764 -id:A053764 - cp's OEIS frontend

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

Ruediger Jehn, Jan 27 2025

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.

			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.

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

Cf. A053764, A007080.

A053766 a(n) = 5^(n^2 - n).

Original entry on oeis.org

1, 1, 25, 15625, 244140625, 95367431640625, 931322574615478515625, 227373675443232059478759765625, 1387778780781445675529539585113525390625

Offset: 0

Stephen G Penrice, Mar 29 2000

Number of nilpotent n X n matrices over GF(5).

N. J. Fine and I. N. Herstein, The probability that a matrix be nilpotent, Illinois J. Math., 2 (1958), 499-504.
M. Gerstenhaber, On the number of nilpotent matrices with coefficients in a finite field. Illinois J. Math., Vol. 5 (1961), 330-333.

Cf. A053763, A053764, A053765.

A053766:=n->5^(n^2 - n); seq(A053766(n), n=0..10); # Wesley Ivan Hurt, Jan 28 2014

Table[5^(n^2 - n), {n, 0, 10}] (* Wesley Ivan Hurt, Jan 28 2014 *)

a(n) = A000351(n^2 - n). - Wesley Ivan Hurt, Jan 28 2014

More terms from James Sellers, Apr 08 2000

A110195 a(n) = 11^((n^2-n)/2).

Original entry on oeis.org

1, 1, 11, 1331, 1771561, 25937424601, 4177248169415651, 7400249944258160101211, 144209936106499234037676064081, 30912680532870672635673352936887453361, 72890483685103052142902866787761839379440139451, 1890591424712781041871514584574319778449301246603238034051

Offset: 0

Philippe Deléham, Sep 07 2005

Sequence given by the Hankel transform (see A001906 for definition) of A082173 = {1, 1, 12, 155, 2124, 30482, 453432, 6936799, ...}; example : det([1, 1, 12, 155; 1, 12, 155, 2124; 12, 155, 2124, 30482; 155, 2124, 30482, 453432]) = 11^6 = 1771561.

Cf. A001020, A006125, A047656, A053763, A053764, A109345, A109354, A109493, A109966, A110147, A161680.

Table[11^((n^2-n)/2),{n,0,20}] (* Harvey P. Dale, Feb 02 2012 *)
Join[{1,1},Table[Det[Table[Binomial[11i,j],{i,n},{j,n}]],{n,10}]] (* Harvey P. Dale, Apr 01 2019 *)

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(11i, j).

a(n) = A001020(A161680(n)).

a(11) from Harvey P. Dale, Feb 02 2012

a(12) from Jason Yuen, Aug 29 2025

cp's OEIS Frontend

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

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A053766 a(n) = 5^(n^2 - n).

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A110195 a(n) = 11^((n^2-n)/2).

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