A338084 Number of equivalence classes of X-based filling of diagonals in a diagonal Latin square of order 2n (or 2n+1).

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

Offset

0,3

Comments

Supplemental for A309283.

The number of solutions in an equivalence class with the main diagonal in ascending order is at most 4*2^n*n!. This maximum is only achieved for n >= 5. - Andrew Howroyd, Mar 27 2023

Links

Table of n, a(n) for n=0..6.

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).

Formula

a(n) >= A000316(n) / (4*2^n*n!). - Andrew Howroyd, Mar 27 2023

Example

From Andrew Howroyd, Mar 27 2023: (Start)
For n = 5, the following is an example solution in an equivalence class of maximum size. The second square shows the effect of swapping the two diagonals and renumbering so that the main diagonal is still in ascending order.
   0 . . . . . . . . 1    0 . . . . . . . . 1
   . 1 . . . . . . 0 .    . 1 . . . . . . 0 .
   . . 2 . . . . 3 . .    . . 2 . . . . 3 . .
   . . . 3 . . 2 . . .    . . . 3 . . 2 . . .
   . . . . 4 6 . . . .    . . . . 4 9 . . . .
   . . . . 7 5 . . . .    . . . . 6 5 . . . .
   . . . 5 . . 6 . . .    . . . 4 . . 6 . . .
   . . 8 . . . . 7 . .    . . 5 . . . . 7 . .
   . 9 . . . . . . 8 .    . 7 . . . . . . 8 .
   4 . . . . . . . . 9    8 . . . . . . . . 9
(End)

Crossrefs

Cf. A000316, A309283.

Sequence in context: A055814 A151370 A041567 * A087301 A292122 A264417

Adjacent sequences: A338081 A338082 A338083 * A338085 A338086 A338087

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, Oct 08 2020

Status

approved