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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). 16
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
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
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-François Alcover, Jul 13 2012, after Jon E. Schoenfield *)
CROSSREFS
Sequence in context: A019589 A087949 A028333 * A208988 A107283 A059237
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

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Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)