A047730 -id:A047730 - 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

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

A047731 Number of score sequences in tournament with n players, when 5 points are awarded in each game.

Original entry on oeis.org

1, 3, 18, 131, 1111, 10461, 105819, 1127413, 12499673, 143021541, 1678718575, 20123155604, 245521479531, 3041006378312, 38157059717410, 484209044329613, 6205758830280388, 80235572611152385

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, A047730.

A064422 Football league numbers: the possible point series 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, 4, 40, 748, 13744, 238568, 4054190

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 of that for chess tournaments (A047730).

			For 2 teams there are 4 possible outcomes: [0, 6], [1, 4], [2, 2] and [3, 3], so a(2) = 4.

Cf. A047730, A064626, A007747.

a(6)-a(7) from Lorand Lucz, Mar 09 2012

A047733 Number of score sequences in tournament with n players, when 6 points are awarded in each game.

Original entry on oeis.org

1, 4, 25, 213, 2131, 23729, 283681, 3574222, 46866712, 634204317, 8803501719, 124799484286, 1800669899917, 26374204955323, 391331674556361, 5872226011836383, 88993282402441857, 1360552594176453319

Offset: 1

David W. Wilson

nonn

Qihao Ye, Table of n, a(n) for n = 1..808

Cf. A000571, A007747, A047729, A047730, A047731.

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}) <= 6, p_i >= 0, i=1, ..., n.

A047735 Number of score sequences in tournament with n players, when 8 points are awarded in each game.

Original entry on oeis.org

1, 5, 41, 459, 6033, 88055, 1379405, 22763356, 390859501, 6924877318, 125837754305, 2335060741480, 44097660919285, 845336236860344, 16415016380975679, 322349248087651458, 6392828942756895663

Offset: 1

David W. Wilson

nonn

Qihao Ye, Table of n, a(n) for n = 1..739

Cf. A000571, A007747, A047729, A047730, A047731.

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}) <= 8, p_i >= 0, i=1, ..., n.

A047736 Number of score sequences in tournament with n players, when 9 points are awarded in each game.

Original entry on oeis.org

1, 5, 50, 630, 9285, 151652, 2658131, 49061128, 942055396, 18662965393, 379195887105, 7867076520341, 166102773740621, 3559787677138284, 77278541685154409, 1696519572528877274

Offset: 1

David W. Wilson

nonn

Qihao Ye, Table of n, a(n) for n = 1..714

Cf. A000571, A007747, A047729, A047730, A047731.

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}) <= 9, p_i >= 0, i=1, ..., n.

A047737 Number of score sequences in tournament with n players, when 10 points are awarded in each game.

Original entry on oeis.org

1, 6, 61, 846, 13771, 248623, 4816659, 98277943, 2086173336, 45688601782, 1026218795502, 23536101285148, 549336702455778, 13014352354398322, 312313455482385108, 7579157833713922471

Offset: 1

David W. Wilson

nonn

Qihao Ye, Table of n, a(n) for n = 1..692

Cf. A000571, A007747, A047729, A047730, A047731.

Nonnegative integer points (p_1, p_2, ..., p_n) in polytope p_0 = p_{n+1} = 0, 2*p_i - (p_{i+1} + p_{i-1}) <= 10, p_i >= 0, i = 1..n.

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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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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A047731 Number of score sequences in tournament with n players, when 5 points are awarded in each game.

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A064422 Football league numbers: the possible point series 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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A047733 Number of score sequences in tournament with n players, when 6 points are awarded in each game.

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A047735 Number of score sequences in tournament with n players, when 8 points are awarded in each game.

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A047736 Number of score sequences in tournament with n players, when 9 points are awarded in each game.

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A047737 Number of score sequences in tournament with n players, when 10 points are awarded in each game.

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