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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). 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 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 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

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Last modified May 9 13:42 EDT 2021. Contains 343742 sequences. (Running on oeis4.)