A309283 - OEIS

A309283

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

1, 1, 0, 0, 2, 2, 3, 3, 20, 20, 67, 67, 596, 596

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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.

K1 = |C[1]|*f[1] + |C[2]|*f[2] + ... + |C[m]|*f[m],

K2 = K1 * n!,

where m = a(n), number of equivalence classes for X-based filling of diagonals in a diagonal Latin square of order n;

C[i], corresponding equivalence classes with cardinalities |C[i]|, 1 <= i <= m;

f[i], the number of diagonal Latin squares corresponds to the each item from equivalence class C[i], 1 <= i <= m;

K1 = A274171(n), number of diagonal Latin squares of order n with fixed first row;

K2 = A274806(n), number of diagonal Latin squares of order n.

For all t>0 a(2*t) = a(2*t+1). - Eduard I. Vatutin, Aug 21 2020

a(14) >= 5225, a(15) >= 5225. - Natalia Makarova, Sep 12 2020

The number of solutions in an equivalence class with the main diagonal in ascending order is at most 4*2^r*r! where r = floor(n/2). This maximum is achieved for orders n >= 10. - Andrew Howroyd, Mar 27 2023

LINKS

Table of n, a(n) for n=0..13.

Stepan Kochemazov, Oleg Zaikin, Eduard Vatutin, and Alexey Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, J. Int. Seq. 23(1) (2020) 20.1.2.

Natalia Makarova, All 67 rules for SN DLS of order 11

Natalia Makarova and Harry White, About unique diagonals for SN DLS of order 14 and 15

Eduard I. Vatutin, About the number of strong normalized lines of diagonal Latin squares of orders 1-10 (in Russian).

Eduard I. Vatutin, About the number of strong normalized lines of diagonal Latin squares of order 11 (in Russian).

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

Eduard I. Vatutin, About the number of equivalence classes of X-based filling of diagonals in a diagonal Latin squares of order 12 (in Russian).

Eduard I. Vatutin, About the number of equivalence classes of X-based filling of diagonals in a diagonal Latin squares of order 13 (in Russian).

Eduard 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).

Oleg Zaikin, Eduard Vatutin, and Curtis Bright, Enumerating Extended Self-Orthogonal Diagonal Latin Squares of Order Up to 10, J. Int. Seq. 28 (2025) 25.7.4. See p. 9.

Index entries for sequences related to Latin squares and rectangles.

FORMULA

a(n) = A338084(floor(n/2)).