A372922 Number of diagonal Latin squares of order 2n+1 that are isomorphic to cyclic Latin squares by row and column permutations.
1, 0, 480, 161280, 2229534720, 45984153600000, 3271798279766016000
Offset: 0
Examples
The cyclic Latin square of order 7 . 0 1 2 3 4 5 6 1 2 3 4 5 6 0 2 3 4 5 6 0 1 3 4 5 6 0 1 2 4 5 6 0 1 2 3 5 6 0 1 2 3 4 6 0 1 2 3 4 5 . has a pair of symmetrically placed transversals T1 = (0, 2, 4, 6, 1, 3, 5) and T2 = (0, 5, 3, 1, 6, 4, 2), after permutting rown and columns transversal T1 placed to the main diagonal with getting single diagonal Latin square . 2 5 0 3 4 6 1 0 3 5 1 2 4 6 1 4 6 2 3 5 0 6 2 4 0 1 3 5 3 6 1 4 5 0 2 4 0 2 5 6 1 3 5 1 3 6 0 2 4 . then after permuting rows and columns transversal T2 placed to the second diagonal with getting diagonal Latin square . 2 5 0 3 6 1 4 0 3 5 1 4 6 2 1 4 6 2 5 0 3 6 2 4 0 3 5 1 4 0 2 5 1 3 6 5 1 3 6 2 4 0 3 6 1 4 0 2 5 . that can be canonized to the following diagonal Latin square: . 0 1 2 3 4 5 6 2 3 1 5 6 4 0 5 6 4 0 1 2 3 4 0 6 2 3 1 5 6 2 0 1 5 3 4 1 5 3 4 0 6 2 3 4 5 6 2 0 1 . Cyclic Latin square of order 11 . 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 0 2 3 4 5 6 7 8 9 10 0 1 3 4 5 6 7 8 9 10 0 1 2 4 5 6 7 8 9 10 0 1 2 3 5 6 7 8 9 10 0 1 2 3 4 6 7 8 9 10 0 1 2 3 4 5 7 8 9 10 0 1 2 3 4 5 6 8 9 10 0 1 2 3 4 5 6 7 9 10 0 1 2 3 4 5 6 7 8 10 0 1 2 3 4 5 6 7 8 9 . can be diagonalized to set of diagonal Latin squares: . 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 10 8 9 0 6 7 1 2 3 4 6 7 8 9 10 0 5 1 2 3 4 5 10 9 0 7 8 6 3 4 5 10 7 9 1 8 2 0 6 8 10 5 7 9 3 0 4 1 6 2 3 4 5 10 6 9 7 2 1 0 8 4 5 10 7 9 6 2 0 3 1 8 4 6 8 10 5 1 7 2 9 3 0 10 6 9 8 7 0 2 5 4 3 1 10 7 9 6 8 0 4 2 5 3 1 9 0 1 2 3 10 4 5 6 7 8 9 8 7 0 1 2 4 6 10 5 3 7 9 6 8 0 1 5 3 10 4 2 7 9 0 1 2 8 3 10 4 5 6 5 10 6 9 8 7 1 4 3 2 0 8 0 1 2 3 4 9 10 6 7 5 6 8 10 5 7 2 9 3 0 4 1 7 0 1 2 3 4 10 8 9 6 5 2 3 4 5 10 7 0 6 1 8 9 10 5 7 9 0 4 1 6 2 8 3 4 5 10 6 9 8 0 3 2 1 7 5 10 7 9 6 8 3 1 4 2 0 3 4 6 8 10 0 5 1 7 2 9 8 7 0 1 2 3 5 9 6 10 4 6 8 0 1 2 3 7 5 9 10 4 2 3 4 6 8 9 10 0 5 1 7 6 9 8 7 0 1 3 10 5 4 2 9 6 8 0 1 2 10 4 7 5 3 5 7 9 0 1 6 2 8 3 10 4 2 3 4 5 10 6 8 1 0 7 9 ... . (totally 81 main classes of diagonal Latin squares).
Links
- Eduard I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
- Eduard I. Vatutin, About the different types of cyclic diagonal Latin squares (in Russian).
- E. Vatutin, A. Belyshev, N. Nikitina, M. Manzuk, A. Albertian, I. Kurochkin, A. Kripachev, and A. Pykhtin, Diagonalization and Canonization of Latin Squares, Lecture Notes in Computer Science, Vol. 14389, Springer, Cham., 2023. pp. 48-61.
- Index entries for sequences related to Latin squares and rectangles.
Formula
a(n) = A372923(n) * (2n+1)!. - Eduard I. Vatutin, Sep 08 2024
Comments