Minimal size of main class for diagonal Latin squares of order n with the first row in order, https://oeis.org/A299783

n=1, a(1)=1
Article: E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586. https://doi.org/10.1007/978-3-030-05807-4_49
Way of finding: brute force
0

n=2, a(2)=0
-

n=3, a(3)=0
-

n=4, a(4)=2
Article: E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586. https://doi.org/10.1007/978-3-030-05807-4_49
Way of finding: brute force
0 1 2 3
2 3 0 1
3 2 1 0
1 0 3 2

n=5, a(5)=4
Article: E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586. https://doi.org/10.1007/978-3-030-05807-4_49
Way of finding: brute force
0 1 2 3 4
2 3 4 0 1
4 0 1 2 3
1 2 3 4 0
3 4 0 1 2

n=6, a(6)=32
Article: E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586. https://doi.org/10.1007/978-3-030-05807-4_49
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=7, a(7)=32
Article: E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586. https://doi.org/10.1007/978-3-030-05807-4_49
Way of finding: brute force
0 1 2 3 4 5 6
1 2 5 6 0 3 4
4 6 3 0 5 1 2
5 4 6 1 3 2 0
3 5 4 2 6 0 1
6 0 1 5 2 4 3
2 3 0 4 1 6 5

n=8, a(8)=96
Article: E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586. https://doi.org/10.1007/978-3-030-05807-4_49
0 1 2 3 4 5 6 7
1 2 3 5 7 6 0 4
3 0 1 7 5 4 2 6
5 6 7 4 3 0 1 2
7 3 5 1 6 2 4 0
4 7 6 0 2 3 5 1
6 5 4 2 0 1 7 3
2 4 0 6 1 7 3 5

n=9, a(9)<=48
Announcement: https://vk.com/wall162891802_1234, Eduard I. Vatutin, Jun 09 2020
Way of finding: one of known orthogonal diagonal Latin squares
0 1 2 3 4 5 6 7 8
2 4 3 0 7 6 8 1 5
6 2 8 5 3 4 7 0 1
4 6 7 1 8 2 3 5 0
1 5 4 7 6 0 2 8 3
7 8 1 4 5 3 0 6 2
3 7 0 2 1 8 5 4 6
8 3 5 6 0 7 1 2 4
5 0 6 8 2 1 4 3 7

n=10, a(10)<=7680
Announcement: https://vk.com/wall162891802_1233, Eduard I. Vatutin, Jun 09 2020
Way of finding: one of known orthogonal diagonal Latin squares
0 1 2 3 4 5 6 7 8 9
1 2 0 4 3 6 5 9 7 8
2 0 3 5 8 1 4 6 9 7
4 6 9 7 1 8 2 0 3 5
9 7 8 6 5 4 3 1 2 0
3 4 7 8 0 9 1 2 5 6
6 9 4 1 7 2 8 5 0 3
7 8 5 0 6 3 9 4 1 2
5 3 1 9 2 7 0 8 6 4
8 5 6 2 9 0 7 3 4 1

n=11, a(11)<=1536
Announcement: https://vk.com/wall162891802_1577, Eduard I. Vatutin, Mar 16 2021
Way of finding: cyclic diagonal Latin squares
0 1 2 3 4 5 6 7 8 9 10
1 2 3 5 6 9 7 10 0 4 8
7 10 8 0 3 1 5 9 6 2 4
3 5 9 4 10 6 8 0 2 7 1
4 6 7 10 1 8 2 3 9 0 5
5 9 4 6 8 7 0 1 3 10 2
10 8 0 1 5 2 9 4 7 3 6
6 7 10 8 2 0 3 5 4 1 9
8 0 1 2 9 3 4 6 10 5 7
2 3 5 9 7 4 10 8 1 6 0
9 4 6 7 0 10 1 2 5 8 3

n=12, a(12)<=46080
Announcement: https://vk.com/wall162891802_1631, Eduard I. Vatutin, Apr 04 2021
Way of finding: search in the neighborhood of central symmetry 
0 1 2 3 4 5 6 7 8 9 10 11 
1 2 0 4 5 3 8 6 7 11 9 10 
3 5 9 0 7 1 10 4 11 2 6 8 
2 10 6 8 11 4 7 0 3 5 1 9 
9 6 1 11 3 7 4 8 0 10 5 2 
8 4 5 2 0 10 1 11 9 6 7 3 
5 9 8 7 10 0 11 1 4 3 2 6 
4 3 11 1 6 2 9 5 10 0 8 7 
7 0 3 5 1 9 2 10 6 8 11 4 
6 8 4 10 2 11 0 9 1 7 3 5 
11 7 10 9 8 6 5 3 2 1 4 0 
10 11 7 6 9 8 3 2 5 4 0 1 

n=13, a(13)<=7680
Announcement: https://vk.com/wall162891802_1578, Eduard I. Vatutin, Mar 16 2021
Way of finding: cyclic diagonal Latin squares
0 1 2 3 4 5 6 7 8 9 10 11 12
2 3 4 6 5 7 9 8 10 11 12 0 1
8 0 10 2 12 1 4 3 6 5 9 7 11
10 2 12 4 1 3 5 6 9 7 11 8 0
6 8 9 10 11 0 12 2 4 1 5 3 7
4 6 5 9 7 8 11 10 12 0 1 2 3
9 10 11 12 0 2 1 4 5 3 7 6 8
12 4 1 5 3 6 7 9 11 8 0 10 2
11 12 0 1 2 4 3 5 7 6 8 9 10
5 9 7 11 8 10 0 12 1 2 3 4 6
7 11 8 0 10 12 2 1 3 4 6 5 9
1 5 3 7 6 9 8 11 0 10 2 12 4
3 7 6 8 9 11 10 0 2 12 4 1 5

Apr 04 2021