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Triangle read by rows: T(n,k) = maximal number of rounds for the social golfer problem with n groups of k golfers (n >= 2, 2 <= k <= n).
3

%I #27 Sep 11 2019 02:43:44

%S 3,5,4,7,4,5,9,7,5,6,11,8,7,6,3,13,10,9

%N Triangle read by rows: T(n,k) = maximal number of rounds for the social golfer problem with n groups of k golfers (n >= 2, 2 <= k <= n).

%C The problem is to find the largest number of rounds of golf that can be arranged with n*k golfers who play in n groups of k. No golfer may play in the same group as any other golfer twice (i.e., maximum socialisation is achieved).

%C T(6,6) cannot be 4 since this would be equivalent to a pair of mutually orthogonal Latin squares of order 6.

%C T(n,k) = 1 for values of n and k outside this range.

%C The next term T(7,5) is known to be either 7 or 8.

%C T(n,n) = A001438(n) + 2. - _Floris P. van Doorn_, Sep 05 2019

%H W. Harvey, <a href="http://web.archive.org/web/20050308115423/http://www.icparc.ic.ac.uk/~wh/golf/">Results Page for the Social Golfer Problem</a>

%H Floris P. van Doorn, <a href="https://github.com/fpvandoorn/Dagstuhl-tables#latin-squares">Dagstuhl's Happy Diner Problem - Latin Squares</a>

%e Triangle begins:

%e 3;

%e 5, 4;

%e 7, 4, 5;

%e 9, 7, 5, 6;

%e 11, 8, 7, 6, 3;

%e ...

%e T(2,2) = 3 from { 12/34, 13/24, 14/23 }.

%Y Column 3 gives A107432.

%Y Cf. A001438.

%K nonn,more,tabl

%O 2,1

%A _N. J. A. Sloane_, following a tip from _Ed Pegg Jr_, May 28 2005