Minimum number of transversals in a diagonal Latin square of order n, https://oeis.org/A287645

n=1, a(1)=1
Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev, Enumerating the Transversals for Diagonal Latin Squares of Small Order. CEUR Workshop Proceedings. Proceedings of the Third International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017. pp. 6-14. urn:nbn:de:0074-1973-0. http://ceur-ws.org/Vol-1973/paper01.pdf
Way of finding: brute force
0

n=2, a(2)=0
-

n=3, a(3)=0
-

n=4, a(4)=8
Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev, Enumerating the Transversals for Diagonal Latin Squares of Small Order. CEUR Workshop Proceedings. Proceedings of the Third International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017. pp. 6-14. urn:nbn:de:0074-1973-0. http://ceur-ws.org/Vol-1973/paper01.pdf
Way of finding: brute force
0 1 2 3
3 2 1 0
1 0 3 2
2 3 0 1

n=5, a(5)=3
Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev, Enumerating the Transversals for Diagonal Latin Squares of Small Order. CEUR Workshop Proceedings. Proceedings of the Third International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017. pp. 6-14. urn:nbn:de:0074-1973-0. http://ceur-ws.org/Vol-1973/paper01.pdf
Way of finding: brute force
0 1 2 3 4
4 2 0 1 3
1 4 3 2 0
3 0 1 4 2
2 3 4 0 1

n=6, a(6)=32
Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev, Enumerating the Transversals for Diagonal Latin Squares of Small Order. CEUR Workshop Proceedings. Proceedings of the Third International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017. pp. 6-14. urn:nbn:de:0074-1973-0. http://ceur-ws.org/Vol-1973/paper01.pdf
Way of finding: brute force
0 1 2 3 4 5
4 2 5 0 3 1
3 5 1 2 0 4
5 3 0 4 1 2
2 4 3 1 5 0
1 0 4 5 2 3

n=7, a(7)=7
Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev, Enumerating the Transversals for Diagonal Latin Squares of Small Order. CEUR Workshop Proceedings. Proceedings of the Third International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017. pp. 6-14. urn:nbn:de:0074-1973-0. http://ceur-ws.org/Vol-1973/paper01.pdf
Way of finding: brute force
0 1 2 3 4 5 6
6 2 3 5 0 1 4
4 5 1 6 2 3 0
2 3 6 4 5 0 1
1 6 5 0 3 4 2
5 0 4 2 1 6 3
3 4 0 1 6 2 5

n=8, a(8)=8
Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev, Enumerating the Transversals for Diagonal Latin Squares of Small Order. CEUR Workshop Proceedings. Proceedings of the Third International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017. pp. 6-14. urn:nbn:de:0074-1973-0. http://ceur-ws.org/Vol-1973/paper01.pdf
Way of finding: brute force
0 1 2 3 4 5 6 7
1 2 0 4 3 7 5 6
4 7 5 1 6 0 2 3
3 5 7 6 1 2 0 4
5 4 6 2 7 1 3 0
7 6 4 0 5 3 1 2
2 3 1 7 0 6 4 5
6 0 3 5 2 4 7 1

n=9, a(9)=68
Announcement: https://vk.com/wall162891802_1347, Eduard I. Vatutin, Natalia N. Nikitina, Maxim O. Manzuk, Sep 17 2020
Way of finding: brute force using X-based fillings
0 1 2 3 4 5 6 7 8
4 8 5 2 1 7 3 0 6
2 5 1 7 0 6 4 8 3
3 2 6 4 8 1 7 5 0
5 3 7 6 2 0 8 1 4
1 6 8 5 7 3 0 4 2
7 4 3 0 6 8 5 2 1
8 7 0 1 3 4 2 6 5
6 0 4 8 5 2 1 3 7

n=10, a(10)<=144
Announcement: https://vk.com/wall162891802_1799, Eduard I. Vatutin, Nov 01 2021
Way of finding: random search in the neighborhood of central symmetry, expanding spectra using 1 intercalate/loop rotation
0 8 9 6 5 2 4 7 3 1 
4 2 6 3 9 0 5 1 7 8 
2 6 5 0 4 8 9 3 1 7 
9 4 7 1 3 5 6 2 8 0 
6 5 1 8 7 9 3 0 2 4 
7 1 3 9 8 4 0 5 6 2 
1 3 0 4 2 7 8 9 5 6 
8 7 2 5 0 3 1 6 4 9 
3 0 8 7 6 1 2 4 9 5 
5 9 4 2 1 6 7 8 0 3 

n=11, a(11)<=1721
Announcement: https://vk.com/wall162891802_1828, Eduard I. Vatutin, Dec 14 2021
Way of finding: random search in the neighborhood of central symmetry + neighborhood based improvement, expanding spectra using 1 intercalate/loop rotation
0 1 2 3 4 5 6 7 8 9 10
1 2 3 4 6 9 10 8 7 0 5
3 10 8 7 0 4 1 9 5 2 6
8 5 9 10 2 1 3 4 6 7 0
5 3 4 6 9 7 8 1 0 10 2
2 6 10 8 7 3 0 5 9 1 4
4 8 7 0 1 6 5 10 2 3 9
7 9 5 2 3 0 4 6 10 8 1
10 4 6 9 8 2 7 0 1 5 3
6 7 0 1 5 10 9 2 3 4 8
9 0 1 5 10 8 2 3 4 6 7

n=12, a(12)<=448
Announcement: https://vk.com/wall162891802_1916, Eduard I. Vatutin, Jan 29 2022
Way of finding: random search in the neighborhood of different types of DLS, combinatorial structures ODLS CFs analysis, expanding transversals and diagonal transversals spectra using rotating of one intercalate/loop
0 1 2 3 4 5 6 7 8 9 10 11
1 2 3 0 8 10 11 9 7 4 6 5
6 11 9 4 1 8 5 3 2 0 7 10
8 7 6 11 10 9 2 1 4 5 0 3
7 9 11 5 6 0 3 2 1 10 8 4
2 3 5 10 7 4 9 11 6 8 1 0
11 5 0 8 2 7 10 4 9 1 3 6
10 6 1 2 5 3 7 8 0 11 4 9
5 10 4 1 11 2 8 0 3 6 9 7
9 0 10 6 3 1 4 5 11 7 2 8
4 8 7 9 0 11 1 6 10 3 5 2
3 4 8 7 9 6 0 10 5 2 11 1

n=13, a(13)<=43093
Announcement: https://vk.com/wall162891802_2161, Eduard I. Vatutin, Jan 09 2023
Way of finding: expanding transversals and diagonal transversals spectra using 1 intercalate/loop rotation, distributed diagonalizing
0 1 2 3 4 5 6 7 8 9 10 11 12
1 2 0 4 3 10 8 11 6 12 5 7 9
8 12 9 10 5 2 4 0 3 11 6 1 7
11 6 7 8 0 4 10 9 2 1 3 12 5
2 5 10 11 7 1 12 3 9 8 0 4 6
9 7 3 12 6 11 1 2 5 0 4 8 10
12 9 5 2 10 6 3 1 4 7 8 0 11
3 4 11 9 12 8 0 6 1 5 7 10 2
5 10 1 7 11 0 9 4 12 6 2 3 8
4 3 12 5 9 7 2 8 0 10 11 6 1
7 11 4 6 8 3 5 12 10 2 1 9 0
6 0 8 1 2 12 7 10 11 4 9 5 3
10 8 6 0 1 9 11 5 7 3 12 2 4

n=14, a(14)<=65432
Announcement: https://vk.com/wall162891802_2016, Eduard I. Vatutin, May 20 2022
Way of finding: Gergely method, expanding spectrum of diagonal transversals using intercalate and loop rotation neighborhoods
0 1 2 3 4 5 6 7 8 9 10 11 12 13
1 2 3 0 9 8 10 12 5 4 6 13 7 11
4 12 5 13 2 1 0 6 11 7 3 8 10 9
10 3 9 1 6 7 8 5 2 0 11 4 13 12
11 13 7 12 8 3 9 4 10 5 1 6 2 0
8 9 6 10 11 13 7 2 3 1 12 0 4 5
9 7 8 11 1 2 3 10 13 12 0 5 6 4
13 11 12 7 5 0 4 9 6 8 2 10 1 3
12 10 13 5 7 9 11 0 4 6 8 2 3 1
7 6 11 8 12 4 13 3 9 10 5 1 0 2
5 4 10 6 13 11 12 1 0 2 7 3 9 8
3 8 1 9 0 6 2 13 7 11 4 12 5 10
6 0 4 2 10 12 5 8 1 3 13 9 11 7
2 5 0 4 3 10 1 11 12 13 9 7 8 6

n=15, a(15)<=215721
Announcement: https://vk.com/wall162891802_1977, Eduard I. Vatutin, Apr 01 2022
Way of finding: expanding spectrum of diagonal transversals using intercalate/loop rotation neighborhoods
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 2 0 4 5 3 7 9 14 6 13 10 8 11 12
10 13 11 14 7 9 2 0 3 1 8 12 5 6 4
9 14 7 8 12 6 4 5 13 3 1 0 10 2 11
4 5 3 10 13 11 8 6 0 12 7 14 2 9 1
8 6 4 2 0 10 11 12 1 13 3 5 9 14 7
5 3 6 13 11 12 14 4 10 8 9 7 0 1 2
2 0 10 5 3 14 9 1 4 7 11 13 6 12 8
7 9 1 6 8 4 3 14 12 5 0 2 11 10 13
14 7 9 12 6 8 5 3 11 4 2 1 13 0 10
12 8 14 1 2 0 13 11 9 10 5 4 7 3 6
11 10 13 9 14 7 1 2 5 0 12 6 4 8 3
13 11 12 7 9 1 0 10 6 2 14 8 3 4 5
3 4 5 11 10 13 12 8 2 14 6 9 1 7 0
6 12 8 0 1 2 10 13 7 11 4 3 14 5 9

n=16, a(16)<=7465984
Announcement: https://vk.com/wall162891802_1903, Eduard I. Vatutin, Jan 23 2022
Way of finding: random Brown DLS, expanding spectrum of diagonal transversals using intercalate/loop rotation neighborhoods
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 0 4 5 11 7 9 14 10 13 12 15 6 3 8
8 13 1 10 12 3 0 6 7 15 11 5 9 14 2 4
5 12 4 9 2 0 11 10 3 13 15 6 7 8 1 14
13 8 14 5 15 9 1 2 0 12 4 10 3 7 6 11
11 4 15 0 14 6 10 3 5 1 9 7 2 12 8 13
10 15 9 2 13 7 3 5 4 6 0 8 14 11 12 1
4 10 3 1 8 14 9 12 2 11 7 13 6 15 5 0
6 14 13 15 9 10 8 0 11 2 5 3 4 1 7 12
14 7 8 11 3 15 13 1 6 4 12 9 10 2 0 5
2 6 7 12 10 4 14 13 1 5 8 15 11 0 9 3
3 9 10 7 1 8 15 11 12 0 6 14 13 5 4 2
7 0 6 8 11 12 2 14 13 3 1 4 5 9 15 10
9 3 5 14 0 13 12 4 15 7 2 1 8 10 11 6
15 11 12 6 7 2 5 8 10 14 3 0 1 4 13 9
12 5 11 13 6 1 4 15 9 8 14 2 0 3 10 7

Jan 09 2023