A337303 Number of X-based filling of diagonals in a diagonal Latin square of order n.

1, 1, 0, 0, 96, 480, 57600, 403200, 191600640, 1724405760, 1597368729600, 17571056025600, 28378507272192000, 368920594538496000, 952903592436341145600, 14293553886545117184000, 55442575636536644075520000, 942523785821122949283840000, 5231730206388249282710863872000

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..100

S. Kochemazov, O. Zaikin, E. Vatutin E., 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, 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) = A337302(n)*n!.

a(n) = n!*A000316(floor(n/2)). - Andrew Howroyd, Mar 26 2023

Example

One of the 96 X-based fillings of diagonals of a diagonal Latin square for order n=4:
1 . . 0
. 0 1 .
. 3 2 .
2 . . 3

Prog

(PARI) \\ here b(n) is A000459.
b(n) = {sum(m=0, n, sum(k=0, n-m, (-1)^k * binomial(n, k) * binomial(n-k, m) * 2^(2*k+m-n) * (2*n-2*m-k)! )); }
a(n) = {2^(n\2) * b(n\2) * n!} \\ Andrew Howroyd, Mar 26 2023

Crossrefs

Cf. A000316, A000459, A309283, A274171, A337302.

Sequence in context: A233710 A234083 A234076 * A216372 A183684 A251167

Adjacent sequences: A337300 A337301 A337302 * A337304 A337305 A337306

Keyword

nonn

Author

Eduard I. Vatutin, Aug 22 2020

Extensions

a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, Mar 26 2023

Status

approved