Minimum number of diagonal transversals in a diagonal Latin square of order n, https://oeis.org/A287647

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)=4
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)=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 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)=2
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)=0
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 4 5 1 0 3
5 0 1 6 2 3 4
3 5 6 4 0 1 2
2 6 5 1 3 4 0
4 3 0 2 5 6 1
1 4 3 0 6 2 5

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

n=9, a(9)=0
Announcement: https://vk.com/wall162891802_1335, Eduard I. Vatutin, Natalia N. Nikitina, Maxim O. Manzuk, Sep 10 2020
Way of finding: brute force using X-based fillings
0 2 3 4 5 7 8 6 1
5 1 6 7 3 8 4 0 2
6 0 2 8 7 1 3 5 4
1 5 7 3 8 2 0 4 6
2 6 1 5 4 0 7 8 3
4 7 8 6 2 5 1 3 0
8 3 5 1 0 4 6 2 7
3 8 4 0 1 6 2 7 5
7 4 0 2 6 3 5 1 8

n=10, a(10)<=3
Announcement: https://vk.com/wall162891802_1796, Eduard I. Vatutin, Nov 02 2021
Way of finding: expanding spectrum of diagonal transversals using loop rotation neighborhoods, diagonalizing
0 1 2 3 4 5 6 7 8 9
1 2 0 4 5 7 9 8 6 3
7 6 4 2 0 9 5 1 3 8
5 9 1 7 8 6 3 2 0 4
9 8 6 1 3 4 2 5 7 0
3 4 9 5 7 8 0 6 2 1
4 7 3 8 6 2 1 0 9 5
8 3 5 6 2 0 4 9 1 7
6 0 8 9 1 3 7 4 5 2
2 5 7 0 9 1 8 3 4 6

n=11, a(11)<=145
Announcement: https://vk.com/wall162891802_1867, Eduard I. Vatutin, Jan 04 2022
Way of finding: random search in the neighborhood of central symmetry, expanding spectrum of diagonal transversals using intercalate/loop rotation neighborhoods, diagonalizing
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)=0
Announcement: https://vk.com/wall162891802_1849, Eduard I. Vatutin, Dec 24 2021
Way of finding: random search in the neighborhood of central symmetry, combinatorial structures ODLS CFs analysis, expanding transversals and diagonal transversals spectra using rotating of one loop, diagonalizing
0 1 2 3 4 5 6 7 8 9 10 11
1 2 0 4 5 7 10 9 6 11 3 8
2 3 5 7 9 11 0 8 10 6 4 1
7 0 4 9 11 8 3 6 2 1 5 10
6 7 8 1 10 2 9 0 5 3 11 4
3 4 7 11 8 6 5 1 0 10 9 2
8 6 10 0 3 4 1 5 11 7 2 9
4 10 3 5 7 9 2 11 1 8 0 6
11 5 9 8 6 1 4 10 3 2 7 0
9 8 1 10 2 0 11 3 7 4 6 5
5 9 11 6 1 10 7 2 4 0 8 3
10 11 6 2 0 3 8 4 9 5 1 7

n=13, a(13)<=4756
Announcement: https://vk.com/wall162891802_2159, 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 5 10 2 3 0 4 7 6 1 11
11 6 7 8 0 3 5 9 2 1 4 12 10
2 10 5 11 7 1 12 4 9 6 0 3 8
9 7 4 12 6 11 1 2 10 0 3 8 5
12 9 10 2 5 6 4 1 3 11 8 0 7
3 4 11 9 12 8 0 6 1 10 7 5 2
10 5 1 7 11 0 9 3 12 8 2 4 6
4 3 12 10 9 7 2 8 0 5 11 6 1
7 11 3 6 8 4 10 12 5 2 1 9 0
6 0 8 1 2 12 7 5 11 3 9 10 4
5 8 6 0 1 9 11 10 7 4 12 2 3

n=14, a(14)<=1446
Announcement: https://vk.com/wall162891802_2849, Eduard I. Vatutin, Oct 09 2024
Way of finding: different source base spectra, expanding spectrum of diagonal transversals using intercalate and loop rotation neighborhoods, partial distributed diagonalizing of the lower part
0 1 2 3 4 5 6 7 8 9 10 11 12 13
1 2 3 0 5 8 9 12 10 11 13 4 6 7
3 0 10 12 6 9 13 8 11 7 4 2 1 5
5 13 6 4 11 0 12 2 1 3 8 9 7 10
4 12 13 6 9 11 7 10 0 5 1 3 2 8
2 6 4 1 13 7 5 11 12 8 3 10 0 9
6 4 1 2 7 12 8 0 3 10 9 13 5 11
8 10 12 11 0 1 3 6 13 4 7 5 9 2
11 8 7 9 3 4 2 13 5 6 0 12 10 1
10 5 11 8 1 13 4 9 7 12 2 0 3 6
12 3 0 10 2 6 1 5 9 13 11 7 8 4
13 7 9 5 8 10 11 3 2 0 6 1 4 12
9 11 8 7 12 3 10 1 4 2 5 6 13 0
7 9 5 13 10 2 0 4 6 1 12 8 11 3

n=15, a(15)<=15510
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, partial diagonalizing
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)<=898988
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

n=17, a(17)<=12058840
Announcement: https://vk.com/wall162891802_1860, Eduard I. Vatutin, Dec 31 2021
Way of finding: random DLS
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 
12 9 0 15 8 4 16 11 5 14 7 10 3 2 6 1 13 
15 6 10 2 3 1 14 12 13 0 5 16 9 11 4 7 8 
2 4 5 16 9 3 8 10 15 7 14 13 1 0 12 11 6 
7 14 3 10 6 2 0 1 9 11 13 5 15 8 16 12 4 
1 0 15 11 16 7 4 2 10 13 6 3 8 12 5 14 9 
3 5 16 6 7 15 13 8 1 12 2 4 10 9 11 0 14 
5 2 12 8 13 9 3 14 7 6 0 15 16 10 1 4 11 
14 12 1 5 15 10 7 16 11 8 3 0 13 4 9 6 2 
16 15 13 9 2 14 11 5 4 1 12 8 6 7 3 10 0 
4 8 14 12 5 16 10 6 2 3 15 9 11 1 0 13 7 
13 16 8 7 10 12 9 4 6 5 11 2 0 14 15 3 1 
9 13 11 1 14 6 15 0 3 10 8 12 4 16 7 2 5 
10 11 6 13 1 0 5 9 12 4 16 14 7 3 2 8 15 
11 3 9 0 12 13 1 15 14 2 4 7 5 6 8 16 10 
6 7 4 14 11 8 12 13 0 16 9 1 2 15 10 5 3 
8 10 7 4 0 11 2 3 16 15 1 6 14 5 13 9 12 

n=18, a(18)<=82577875
Announcement: https://vk.com/wall162891802_1860, Eduard I. Vatutin, Dec 31 2021
Way of finding: random DLS
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 
5 7 13 15 8 17 11 2 6 0 14 10 16 3 9 4 12 1 
6 13 1 4 9 15 8 10 2 5 3 0 7 17 16 11 14 12 
3 14 5 11 15 6 12 9 7 1 2 16 10 0 13 8 17 4 
11 8 0 5 17 13 14 4 3 6 12 2 9 1 15 7 10 16 
12 0 7 17 3 14 4 13 1 15 11 5 8 9 6 16 2 10 
16 17 15 7 10 12 5 3 14 13 8 9 4 2 11 1 6 0 
8 11 3 13 12 4 10 16 9 14 7 15 0 5 2 17 1 6 
14 3 12 16 11 1 15 6 4 2 13 17 5 7 8 10 0 9 
10 4 9 12 5 16 13 14 15 3 1 6 2 11 17 0 8 7 
1 2 4 14 0 10 17 5 12 11 15 8 6 16 3 9 7 13 
15 9 10 8 14 11 16 17 0 7 4 12 3 6 1 2 13 5 
2 12 17 1 7 3 9 8 16 4 6 14 13 10 0 5 11 15 
7 16 11 2 6 9 3 12 17 10 0 1 15 8 5 13 4 14 
13 6 16 0 1 7 2 11 5 17 9 3 14 4 10 12 15 8 
9 15 14 10 13 2 0 1 11 8 16 7 17 12 4 6 5 3 
17 10 8 6 2 0 7 15 13 16 5 4 1 14 12 3 9 11 
4 5 6 9 16 8 1 0 10 12 17 13 11 15 7 14 3 2 

n=19, a(19)<=592174879
Announcement: https://vk.com/wall162891802_1861, Eduard I. Vatutin, Dec 31 2021
Way of finding: random DLS
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
9 14 3 17 16 0 12 1 13 7 2 4 6 8 18 10 15 5 11
18 16 6 0 8 15 5 14 3 4 9 7 17 10 12 2 13 11 1
2 18 1 12 14 10 3 15 9 8 4 17 5 6 11 16 0 13 7
6 5 16 18 1 11 9 13 10 17 14 0 7 3 15 12 8 4 2
16 12 13 5 11 2 4 9 6 0 7 8 15 14 1 3 10 18 17
4 9 11 16 6 3 15 12 18 5 0 13 8 7 2 14 17 1 10
1 7 10 9 5 18 13 17 15 3 16 6 14 4 8 0 11 2 12
15 11 12 6 3 7 2 10 5 18 1 14 9 16 13 17 4 0 8
13 0 17 4 12 8 11 6 2 10 5 18 3 9 7 1 14 16 15
11 13 14 15 0 17 7 3 12 1 18 10 16 5 6 9 2 8 4
7 4 8 10 18 13 0 11 1 6 17 16 2 15 3 5 12 9 14
5 3 9 1 2 14 17 8 11 12 6 15 13 0 10 4 18 7 16
12 10 7 13 17 4 8 2 14 15 3 9 1 11 16 18 5 6 0
17 6 5 2 9 1 16 0 7 14 8 12 10 18 4 11 3 15 13
14 17 18 7 13 12 1 4 16 11 15 5 0 2 9 8 6 10 3
8 15 0 11 10 9 18 16 17 2 12 3 4 1 5 13 7 14 6
10 2 4 8 15 6 14 18 0 16 13 1 11 12 17 7 9 3 5
3 8 15 14 7 16 10 5 4 13 11 2 18 17 0 6 1 12 9

n=20, a(20)<=4488686380
Announcement: https://vk.com/wall162891802_19511, Eduard I. Vatutin, Feb 19 2022
Way of finding: random centrally symmetric DLS
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
17 4 14 7 6 3 12 13 1 0 19 10 9 18 15 11 5 8 16 2
18 13 10 4 19 6 3 17 11 16 8 1 14 15 0 12 2 9 7 5
8 5 6 19 17 11 9 1 0 2 4 12 15 7 10 16 18 3 14 13
19 9 5 15 1 2 18 8 7 3 16 6 11 4 13 14 0 12 10 17
13 0 18 16 10 17 7 19 9 15 11 14 8 6 4 1 3 5 2 12
9 7 1 18 11 0 15 3 10 14 13 8 2 5 12 19 6 4 17 16
5 16 7 14 0 19 13 11 17 10 3 2 6 12 8 4 9 1 15 18
15 2 9 8 18 13 4 10 12 11 7 17 16 3 19 0 1 14 5 6
4 8 19 5 2 12 0 9 14 6 1 7 10 11 18 17 13 16 3 15
14 17 12 1 3 10 5 18 19 13 2 15 0 9 16 6 11 7 4 8
2 11 15 13 9 7 17 12 3 19 5 16 18 8 1 10 4 0 6 14
10 14 13 0 8 4 16 2 6 17 18 3 5 1 7 9 15 19 12 11
12 3 8 2 7 16 11 6 4 1 15 18 17 14 9 5 10 13 19 0
16 6 11 17 13 8 2 4 15 5 14 0 7 19 3 18 12 10 9 1
3 18 16 9 15 1 8 5 2 12 17 19 4 10 6 13 14 11 0 7
1 15 17 10 12 18 19 14 16 8 6 9 13 0 5 3 7 2 11 4
11 19 0 6 16 9 14 15 13 4 12 5 3 17 2 7 8 18 1 10
6 12 4 11 5 14 10 0 18 7 9 13 1 16 17 2 19 15 8 3
7 10 3 12 14 15 1 16 5 18 0 4 19 2 11 8 17 6 13 9

Oct 09 2024