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.
Sophie Rehberg, Extensions of Ehrhart theory and applications to combinatorial structures, Ph D. dissertation, Freien Univ. (Berlin, Germany 2025). See pp. 8, 120, 144.
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
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