Maximum number of diagonal 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)=4
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)=6
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 5 1 4 0 2 
5 4 3 2 1 0 
2 0 4 1 5 3 

n=8, a(8)=120
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 
2 3 0 1 6 7 4 5 
1 5 4 0 7 3 2 6 
5 4 7 6 1 0 3 2 
3 7 6 2 5 1 0 4 
7 6 5 4 3 2 1 0 
4 0 1 5 2 6 7 3 
6 2 3 7 0 4 5 1 

n=10, a(10)>=890
Announcement: https://vk.com/wall162891802_1794, Eduard I. Vatutin, Nov 01 2021
Way of finding: expanding spectrum of diagonal transversals of DLS using loop rotation neighborhoods, diagonalizing
0 1 2 3 4 5 6 7 8 9 
1 2 3 4 0 9 5 6 7 8 
3 4 9 8 2 7 1 0 5 6 
6 5 0 1 7 2 8 9 4 3 
9 8 7 6 5 4 3 2 1 0 
4 0 8 2 3 6 7 1 9 5 
8 7 6 5 9 0 4 3 2 1 
5 9 1 7 6 3 2 8 0 4 
7 6 5 9 8 1 0 4 3 2 
2 3 4 0 1 8 9 5 6 7 

n=12, a(12)>=28496
Announcement: -, Eduard I. Vatutin, ~Aug 18 2021
Way of finding: diagonalizing of known double Brown square (obtained using the composite squares method + diagonalizing)
0 1 2 3 4 5 6 7 8 9 10 11 
1 2 3 4 5 0 11 6 7 8 9 10 
5 0 1 2 3 4 7 8 9 10 11 6 
8 7 6 11 10 9 2 1 0 5 4 3 
7 6 11 10 9 8 3 2 1 0 5 4 
9 8 7 6 11 10 1 0 5 4 3 2 
6 11 10 9 8 7 4 3 2 1 0 5 
10 9 8 7 6 11 0 5 4 3 2 1 
11 10 9 8 7 6 5 4 3 2 1 0 
2 3 4 5 0 1 10 11 6 7 8 9 
4 5 0 1 2 3 8 9 10 11 6 7 
3 4 5 0 1 2 9 10 11 6 7 8 

n=14, a(14)>=490218
Announcement: https://vk.com/wall162891802_2440, Eduard I. Vatutin, Jul 16 2023
Way of finding: full distributed diagonalizing of top (by transversals number) known DLS
0 1 2 3 4 5 6 7 8 9 10 11 12 13
1 2 3 4 5 6 0 13 7 8 9 10 11 12
3 4 5 6 0 1 2 11 12 13 7 8 9 10
11 10 9 8 7 13 12 1 0 6 5 4 3 2
9 8 7 13 12 11 10 3 2 1 0 6 5 4
8 7 13 12 11 3 9 4 10 2 1 0 6 5
12 11 10 9 8 7 13 0 6 5 4 3 2 1
5 6 0 1 2 10 4 9 3 11 12 13 7 8
6 0 1 2 10 9 5 8 4 3 11 12 13 7
7 13 12 11 3 4 8 5 9 10 2 1 0 6
10 9 8 7 13 12 11 2 1 0 6 5 4 3
4 5 6 0 1 2 3 10 11 12 13 7 8 9
13 12 11 10 9 8 7 6 5 4 3 2 1 0
2 3 4 5 6 0 1 12 13 7 8 9 10 11

n=16, a(16)>=32172800
Announcement: https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=109&postid=1201, Natalia Makarova, Jan 15 2021
Way of finding: composite squares method
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12
1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14
2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13
12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2
14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1
4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11
7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8
5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10
6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9
8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7
11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4
9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6
10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5

Dec 22 2024