A337302 - OEIS

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

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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 .