|
%I #31 Jul 08 2023 19:24:25
%S 1,2,5,16,59,247,1111,5302,26376,135670,716542,3868142,21265884,
%T 118741369,671906876,3846342253,22243294360,129793088770,763444949789,
%U 4522896682789,26968749517543,161750625450884
%N 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).
%C A correspondence between the points in the polytope and the chess scores was found by Svante Linusson (linusson(AT)matematik.su.se):
%C 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.
%D P. A. MacMahon, Chess tournaments and the like treated by the calculus of symmetric functions, Coll. Papers I, MIT Press, 344-375.
%H Qihao Ye, <a href="/A007747/b007747.txt">Table of n, a(n) for n = 0..1214</a> (terms 0..39 from Jon E. Schoenfield)
%H P. Di Francesco, M. Gaudin, C. Itzykson and F. Lesage, <a href="http://dx.doi.org/10.1142/S0217751X94001734">Laughlin's wave functions, Coulomb gases and expansions of the discriminant</a>, Int. J. Mod. Phys. A9 (1994) 4257.
%H Jon E. Schoenfield, <a href="/A007747/a007747.txt">Comments on this sequence</a>
%H <a href="/index/To#tournament">Index entries for sequences related to tournaments</a>
%F Schoenfield (see Comments link) gives a recursive method for computing this sequence.
%e With 3 players the possible scores sequences are {{0,2,4}, {0,3,3}, {1,1,4}, {1,2,3}, {2,2,2}}.
%e 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}}.
%t 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_ *)
%Y Cf. A000571, A047730, A064626, A064422.
%K nonn,nice
%O 0,2
%A P. Di Francesco (philippe(AT)amoco.saclay.cea.fr), _N. J. A. Sloane_
%E More terms from _David W. Wilson_
|
