Maximum number of transversals in Brown's diagonal Latin squares of order N=2n, https://oeis.org/Axxxxxx

n=2, a(2)=0
Announcement: https://vk.com/wall162891802_2895, Eduard I. Vatutin, Dec 16 2024
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n=4, a(4)=8
Announcement: https://vk.com/wall162891802_2895, Eduard I. Vatutin, Dec 16 2024
Way of finding: brute force
0 1 2 3 
2 3 0 1 
3 2 1 0 
1 0 3 2 

n=6, a(6)=32
Announcement: https://vk.com/wall162891802_2895, Eduard I. Vatutin, Dec 16 2024
Way of finding: brute force
0 1 2 3 4 5 
1 2 0 5 3 4 
4 3 5 0 2 1 
3 0 1 4 5 2 
5 4 3 2 1 0 
2 5 4 1 0 3 

n=8, a(8)=384
Announcement: https://vk.com/wall162891802_2895, Eduard I. Vatutin, Dec 16 2024
Way of finding: brute force
0 1 2 3 4 5 6 7 
1 2 3 0 7 4 5 6 
3 7 1 5 2 6 0 4 
7 6 5 4 3 2 1 0 
5 3 7 1 6 0 4 2 
6 5 4 7 0 3 2 1 
4 0 6 2 5 1 7 3 
2 4 0 6 1 7 3 5 

n=10, a(10)>=5504
Article: J. W. Brown, F. Cherry, L. Most, M. Most, E. T. Parker, W. D. Wallis, Completion of the spectrum of orthogonal diagonal Latin squares, Lecture notes in pure and applied mathematics, 1992, Vol. 139, pp. 43-49.
Way of finding: ?
0 8 5 1 7 3 4 6 9 2 
5 1 7 2 9 8 0 3 4 6 
1 7 2 9 5 6 8 0 3 4 
9 6 4 3 0 2 7 1 5 8 
3 0 8 6 4 1 5 9 2 7 
4 3 0 8 6 5 9 2 7 1 
7 2 9 5 1 4 6 8 0 3 
6 4 3 0 8 9 2 7 1 5 
2 9 6 4 3 7 1 5 8 0 
8 5 1 7 2 0 3 4 6 9 

n=12, a(12)>=198144
Announcement: https://vk.com/wall162891802_1603, Eduard I. Vatutin, Mar 25 2021 (was known before as LS, see https://users.monash.edu.au/~iwanless/data/transversals/)
Way of finding: composite squares method + diagonalizing/rotating subsquares
0 1 2 3 4 5 6 7 8 9 10 11 
1 2 3 4 5 0 11 6 7 8 9 10 
9 8 7 6 11 10 1 0 5 4 3 2 
4 5 0 1 2 3 8 9 10 11 6 7 
6 11 10 9 8 7 4 3 2 1 0 5 
11 10 9 8 7 6 5 4 3 2 1 0 
3 4 5 0 1 2 9 10 11 6 7 8 
2 3 4 5 0 1 10 11 6 7 8 9 
10 9 8 7 6 11 0 5 4 3 2 1 
5 0 1 2 3 4 7 8 9 10 11 6 
7 6 11 10 9 8 3 2 1 0 5 4 
8 7 6 11 10 9 2 1 0 5 4 3 

n=14, a(14)>=3477504
Announcement: https://vk.com/wall162891802_1914, Eduard I. Vatutin, Jan 27 2022
Way of finding: expanding spectrum of diagonal transversals in DLS using intercalate rotation neighborhoods
0 1 2 3 9 5 7 6 8 4 10 11 12 13
13 12 11 10 4 8 6 7 5 9 3 2 1 0
10 2 5 6 12 13 9 4 0 1 7 8 11 3
1 3 6 11 0 9 8 5 4 13 2 7 10 12
9 0 3 1 8 6 11 2 7 5 12 10 13 4
5 9 12 0 6 2 3 10 11 7 13 1 4 8
11 7 9 8 3 1 13 0 12 10 5 4 6 2
3 11 8 7 1 0 4 9 13 12 6 5 2 10
7 8 0 4 11 3 12 1 10 2 9 13 5 6
8 4 1 13 7 11 10 3 2 6 0 12 9 5
6 5 13 9 2 10 1 12 3 11 4 0 8 7
4 13 10 12 5 7 2 11 6 8 1 3 0 9
2 6 4 5 10 12 0 13 1 3 8 9 7 11
12 10 7 2 13 4 5 8 9 0 11 6 3 1

n=16, a(16)>=244744192
Announcement 1 (value without proving DLS): https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=138&postid=2731, Natalia Makarova, Jul 30 2021 (was known before as LS, see https://users.monash.edu.au/~iwanless/data/transversals/)
Announcement 2 (value and DLS): https://vk.com/wall162891802_1900, Eduard I. Vatutin, Jan 22 2022
Way of finding: composite squares method
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 3 0 1 5 4 7 6 9 8 11 10 14 15 12 13
3 2 1 0 6 7 4 5 10 11 8 9 15 14 13 12
1 0 3 2 7 6 5 4 11 10 9 8 13 12 15 14
8 11 9 10 14 12 13 15 0 2 3 1 5 6 4 7
10 9 11 8 13 15 14 12 3 1 0 2 7 4 6 5
11 8 10 9 15 13 12 14 1 3 2 0 6 5 7 4
9 10 8 11 12 14 15 13 2 0 1 3 4 7 5 6
14 15 12 13 8 9 10 11 4 5 6 7 2 3 0 1
13 12 15 14 10 11 8 9 6 7 4 5 1 0 3 2
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
12 13 14 15 9 8 11 10 5 4 7 6 0 1 2 3
6 5 7 4 3 1 0 2 13 15 14 12 11 8 10 9
7 4 6 5 1 3 2 0 15 13 12 14 10 9 11 8
4 7 5 6 0 2 3 1 14 12 13 15 9 10 8 11
5 6 4 7 2 0 1 3 12 14 15 13 8 11 9 10

Dec 16 2024