A309985 - OEIS

%I #68 Dec 12 2020 06:31:38

%S 1,1,3,18,160,2325,41895,961772,26978400,929587995

%N Maximum determinant of an n X n Latin square.

%C a(n) = A301371(n) for n <= 7. a(8) < A301371(8) = 27296640, a(9) < A301371(9) = 933251220.

%C a(10) = 36843728625, conjectured. See Stack Exchange link. - _Hugo Pfoertner_, Sep 29 2019

%C A328030(n) <= a(n) <= A301371(n). - _Hugo Pfoertner_, Dec 02 2019

%C It is unknown, but very likely, that A301371(n) > a(n) also holds for all n > 9 - _Hugo Pfoertner_, Dec 12 2020

%H Brendan McKay, <a href="https://users.cecs.anu.edu.au/~bdm/data/latin.html">Latin squares</a>.

%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/885481/maximum-determinant-of-latin-squares">Maximum determinant of Latin squares</a>, (2014), (2016).

%e An example of an 8 X 8 Latin square with maximum determinant is

%e   [7  1  3  4  8  2  5  6]

%e   [1  7  4  3  6  5  2  8]

%e   [3  4  1  7  2  6  8  5]

%e   [4  3  7  1  5  8  6  2]

%e   [8  6  2  5  4  7  1  3]

%e   [2  5  6  8  7  3  4  1]

%e   [5  2  8  6  1  4  3  7]

%e   [6  8  5  2  3  1  7  4].

%e An example of a 9 X 9 Latin square with maximum determinant is

%e   [9  4  3  8  1  5  2  6  7]

%e   [3  9  8  5  4  6  1  7  2]

%e   [4  1  9  3  2  8  7  5  6]

%e   [1  2  4  9  7  3  6  8  5]

%e   [8  3  5  6  9  7  4  2  1]

%e   [2  7  1  4  6  9  5  3  8]

%e   [5  8  6  7  3  2  9  1  4]

%e   [7  6  2  1  5  4  8  9  3]

%e   [6  5  7  2  8  1  3  4  9].

%e An example of a 10 X 10 Latin square with abs(determinant) = 36843728625 is a circulant matrix with first row [1, 3, 7, 9, 8, 6, 5, 4, 2, 10], but it is not known if this is the best possible. - _Kebbaj Mohamed Reda_, Nov 27 2019 (reworded by _Hugo Pfoertner_)

%Y Cf. A040082, A301371, A308853, A309258, A309984, A328029, A328030.

%K nonn,hard,more

%O 0,3

%A _Hugo Pfoertner_, Aug 26 2019

%E a(9) from _Hugo Pfoertner_, Aug 30 2019

%E a(0)=1 prepended by _Alois P. Heinz_, Oct 02 2019