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

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.

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.

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

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

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

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

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

E. I. Vatutin, About the number of equivalence classes of X-based filling of diagonals in a diagonal Latin squares of order 13 (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) = A338084(floor(n/2)).

Example

For order n=4 there are a(4)=2 equivalence classes. First of them C[1] includes two X-based fillings of diagonals
   0..1  0..2
   .13.  .10.
   .02.  .32.
   2..3  1..3
and second C[2] also includes two X-based fillings of diagonals
   0..1  0..2
   .10.  .13.
   .32.  .02.
   2..3  1..3
It is easy to see that f[1] = 0 and f[2] = 1, so K1(4) = A274171(4) = 2*0 + 2*1 = 2 and K2(4) = A274806(4) = K1(4) * 4! = 2 * 24 = 48.

Crossrefs

Cf. A274171, A274806, A337302, A338084.

Sequence in context: A216245 A180000 A053094 * A196080 A124516 A205958

Adjacent sequences: A309280 A309281 A309282 * A309284 A309285 A309286

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, Jul 06 2020

Extensions

a(11) added by Eduard I. Vatutin, Aug 21 2020

a(12)-a(13) by Harry White, added by Natalia Makarova, Sep 12 2020

a(0)=1 prepended by Andrew Howroyd, Oct 31 2020

Status

approved