A064422 -id:A064422 - cp's OEIS frontend

A007747 Number of nonnegative integer points (p_1,p_2,...,p_n) in polytope defined by p_0 = p_{n+1} = 0, 2p_i - (p_{i+1} + p_{i-1}) <= 2, p_i >= 0, i=1,...,n. Number of score sequences in a chess tournament with n+1 players (with 3 outcomes for each game).

1, 2, 5, 16, 59, 247, 1111, 5302, 26376, 135670, 716542, 3868142, 21265884, 118741369, 671906876, 3846342253, 22243294360, 129793088770, 763444949789, 4522896682789, 26968749517543, 161750625450884

Offset: 0

P. Di Francesco (philippe(AT)amoco.saclay.cea.fr), N. J. A. Sloane

A correspondence between the points in the polytope and the chess scores was found by Svante Linusson (linusson(AT)matematik.su.se):

The score sequences are partitions (a_1,...,a_n) of 2C(n,2) of length <= n that are majorized by 2n,2n-2,2n-4,...,2,0; i.e. f(n,k) := 2n+2n-2+...+(2n-2k+2)-(a_1+a_2+...+a_k) >= 0 for all k. The sequence 0=f(n,0),f(n,1),f(n,2),...,f(n,n)=0 is in the polytope. This establishes the bijection.

			With 3 players the possible scores sequences are {{0,2,4}, {0,3,3}, {1,1,4}, {1,2,3}, {2,2,2}}.
With 4 players they are {{0,2,4,6}, {0,2,5,5}, {0,3,3,6}, {0,3,4,5}, {0,4,4,4}, {1,1,4,6}, {1,1,5,5}, {1,2,3,6}, {1,2,4,5}, {1,3,3,5}, {1,3,4,4}, {2,2,2,6}, {2,2,3,5}, {2,2,4,4}, {2,3,3,4}, {3,3,3,3}}.

P. A. MacMahon, Chess tournaments and the like treated by the calculus of symmetric functions, Coll. Papers I, MIT Press, 344-375.

Qihao Ye, Table of n, a(n) for n = 0..1214 (terms 0..39 from Jon E. Schoenfield)
P. Di Francesco, M. Gaudin, C. Itzykson and F. Lesage, Laughlin's wave functions, Coulomb gases and expansions of the discriminant, Int. J. Mod. Phys. A9 (1994) 4257.
Jon E. Schoenfield, Comments on this sequence
Index entries for sequences related to tournaments

Cf. A000571, A047730, A064626, A064422.

f[K_, L_, S_, X_] /; K > 1 && L <= S/K <= X + 1 - K := f[K, L, S, X] = Sum[f[K - 1, i, S - i, X], {i, L, Floor[S/K]}]; f[1, L_, S_, X_] /; L <= S <= X = 1; f[, , , ] = 0; a[n_] := f[n + 1, 0, n*(n + 1), 2*n]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 13 2012, after Jon E. Schoenfield *)

Schoenfield (see Comments link) gives a recursive method for computing this sequence.

More terms from David W. Wilson

A047730 Number of score sequences in tournament with n players, when 4 points are awarded in each game.

Original entry on oeis.org

1, 3, 13, 76, 521, 3996, 32923, 286202, 2590347, 24203935, 232050202, 2272449745, 22653570386, 229274897514, 2350933487206, 24381053759852, 255382755251622, 2698732882975782, 28743579211912338

Offset: 1

David W. Wilson

nonn

P. A. MacMahon, Chess tournaments and the like treated by the calculus o symmetric functions, Coll. Papers I, MIT Press, 344-375.

Cf. A000571, A007747, A047729, A064626, A064422.

Nonnegative integer points (p_1, p_2, ..., p_n) in polytope p_0=p_{n+1}=0, 2p_i -(p_{i+1}+p_{i-1}) <= 4, p_i >= 0, i=1, ..., n.

A064626 Football tournament numbers: the number of possible point series for a tournament of n teams playing each other once where 3 points are awarded to the winning team and 1 to each in the case of a tie.

Original entry on oeis.org

1, 2, 7, 40, 355, 3678, 37263, 361058, 3403613, 31653377, 292547199, 2696619716

Offset: 1

Thomas Schulze (jazariel(AT)tiscalenet.it), Sep 30 2001

This sequence reflects the now common 3-point rule of international football where the sum of total points awarded depends on the outcome of each match. The classical 2-point rule is equivalent to that for chess tournaments (A007747).

			For 2 teams there are 2 possible outcomes: [0, 3] and [1, 1], so a(2) = 2.
For 3 teams the outcomes are [0, 3, 6], [1, 3, 4], [3, 3, 3], [1, 1, 6], [1, 2, 4], [0, 4, 4] and [2, 2, 2], so a(3) is 7. Note that the outcome [3, 3, 3] can be obtained in two ways: (A beats B, B beats C, C beats A) or (B beats A, A beats C, C beats B).

A. Ivanyi, L. Lucz, T. Matuszka, and S. Pirzada, Parallel enumeration of degree sequences of simple graphs, Acta Univ. Sapientiae, Informatica, 4, 2 (2012) 260-288. - From _N. J. A. Sloane_, Feb 15 2013
Wikipedia, Three points for a win

Cf. A007747, A047730, A064422, A152789.

a(8) and a(9) from Jon E. Schoenfield, May 05 2007
a(10) from Ming Li (dawnli(AT)ustc.edu), Jun 20 2008
a(11) from Jon E. Schoenfield, Sep 04 2008
a(12) from Jon E. Schoenfield, Dec 12 2008

A209467 Football league numbers with distinct point totals for a league of n teams playing each other twice where for each match 3 points are awarded to the winning team and 1 to each in the case of a tie.

Original entry on oeis.org

1, 2, 25, 408, 6272, 91640, 1316363

Offset: 1

Lorand Lucz, Mar 09 2012

nonn

			For n=2 the possible outcomes are [0,6] and [1,4], so a(2)=2.  The other possible outcomes [2,2] and [3,3] do not have distinct point totals.

A064422 allows teams to have the same point total.

Cf. A064626, A152789

cp's OEIS Frontend

A007747 Number of nonnegative integer points (p_1,p_2,...,p_n) in polytope defined by p_0 = p_{n+1} = 0, 2p_i - (p_{i+1} + p_{i-1}) <= 2, p_i >= 0, i=1,...,n. Number of score sequences in a chess tournament with n+1 players (with 3 outcomes for each game).

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A047730 Number of score sequences in tournament with n players, when 4 points are awarded in each game.

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A064626 Football tournament numbers: the number of possible point series for a tournament of n teams playing each other once where 3 points are awarded to the winning team and 1 to each in the case of a tie.

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A209467 Football league numbers with distinct point totals for a league of n teams playing each other twice where for each match 3 points are awarded to the winning team and 1 to each in the case of a tie.

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Author

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Examples

Crossrefs