A337302 Number of X-based filling of diagonals in a diagonal Latin square of order n with the main diagonal in ascending order.

1, 1, 0, 0, 4, 4, 80, 80, 4752, 4752, 440192, 440192, 59245120, 59245120, 10930514688, 10930514688, 2649865335040, 2649865335040, 817154768973824, 817154768973824, 312426715251262464, 312426715251262464, 145060238642780180480, 145060238642780180480

Offset

0,5

Comments

Used for getting strong canonical forms (SCFs) of the diagonal Latin squares and for fast enumerating of the diagonal Latin squares based on equivalence classes.

For all t > 0, a(2*t) = a(2*t+1).

Links

Table of n, a(n) for n=0..23.

S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.

E. I. Vatutin, About the number of X-based fillings of diagonals in a diagonal Latin squares of orders 1-15 (in Russian).

E. I. Vatutin, About the a(2*t)=a(2*t+1) equality (in Russian).

E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).

Index entries for sequences related to Latin squares and rectangles.

Formula

a(n) = A337303(n)/n!.

a(n) = A000316(floor(n/2)). - Andrew Howroyd and Eduard I. Vatutin, Oct 08 2020

Example

For n=4 there are 4 different X-based fillings of diagonals with main diagonal fixed to [0 1 2 3]:
   0 . . 1   0 . . 1   0 . . 2   0 . . 2
   . 1 0 .   . 1 3 .   . 1 0 .   . 1 3 .
   . 3 2 .   . 0 2 .   . 3 2 .   . 0 2 .
   2 . . 3   2 . . 3   1 . . 3   1 . . 3

Crossrefs

Cf. A000316, A309283, A274171, A337303.

Sequence in context: A222426 A107053 A327303 * A351349 A222271 A068376

Adjacent sequences: A337299 A337300 A337301 * A337303 A337304 A337305

Keyword

nonn

Author

Eduard I. Vatutin, Aug 22 2020

Extensions

More terms from Alois P. Heinz, Oct 08 2020

a(0)=1 prepended by Andrew Howroyd, Oct 09 2020

Status

approved