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Search: id:A007747
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| A007747 |
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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). |
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+0
14
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| 1, 2, 5, 16, 59, 247, 1111, 5302, 26376, 135670, 716542, 3868142, 21265884, 118741369, 671906876, 3846342253, 22243294360, 129793088770, 763444949789, 4522896682789, 26968749517543, 161750625450884
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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 majorised 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.
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REFERENCES
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P. Di Francesco, M. Gaudin, C. Itzykson and F. Lesage, Laughlin's wave functions, Coulomb gases and expansions of the discriminant, Int. Jour. of Mod. Phys. A, Vol. 9, No. 24 (1994) 4257-4351.
P. A. MacMahon, Chess tournaments and the like treated by the calculus of symmetric functions, Coll. Papers I, MIT Press, 344-375.
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LINKS
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Jon Schoenfield, Table of n, a(n) for n = 0..39
Jon Schoenfield, Comments on this sequence
Index entries for sequences related to tournaments
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FORMULA
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Schoenfield (see Comments link) gives a recursive method for computing this sequence.
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EXAMPLE
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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}}.
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CROSSREFS
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Cf. A000571, A047730, A064626, A064422.
Sequence in context: A019589 A087949 A028333 this_sequence A107283 A059237 A104547
Adjacent sequences: A007744 A007745 A007746 this_sequence A007748 A007749 A007750
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KEYWORD
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nonn,nice
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AUTHOR
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P. Di Francesco (philippe(AT)amoco.saclay.cea.fr), N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from David W. Wilson (davidwwilson(AT)comcast.net)
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