

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_{i1}) <= 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
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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,2n2,2n4,...,2,0; i.e. f(n,k) := 2n+2n2+...+(2n2k+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, 344375.


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}] (* JeanFrançois 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



