login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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). 15
1, 2, 5, 16, 59, 247, 1111, 5302, 26376, 135670, 716542, 3868142, 21265884, 118741369, 671906876, 3846342253, 22243294360, 129793088770, 763444949789, 4522896682789, 26968749517543, 161750625450884 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

REFERENCES

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

LINKS

Jon E. Schoenfield, Table of n, a(n) for n = 0..39

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

FORMULA

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

EXAMPLE

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}}.

MATHEMATICA

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-Fran├žois Alcover, Jul 13 2012, after Jon E. Schoenfield *)

CROSSREFS

Cf. A000571, A047730, A064626, A064422.

Sequence in context: A019589 A087949 A028333 * A208988 A107283 A059237

Adjacent sequences:  A007744 A007745 A007746 * A007748 A007749 A007750

KEYWORD

nonn,nice

AUTHOR

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

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 18 12:18 EDT 2019. Contains 328160 sequences. (Running on oeis4.)