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A293777 Number of centrally symmetric diagonal Latin squares of order n with the first row in ascending order. 4
1, 0, 0, 2, 8, 0, 2816, 135168, 327254016 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

A centrally symmetric diagonal Latin square is a square with one-to-one correspondence between elements within all pairs a[i][j] and a[n-1-i][n-1-j] (with rows and columns numbered from 0 to n-1).

a(n)=0 for n=2 and n=3 (diagonal Latin squares of these sizes don't exist). It seems that a(n)=0 for n == 2 (mod 4).

Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(4n) >= A287650(n). - Eduard I. Vatutin, May 03 2021

LINKS

Table of n, a(n) for n=1..9.

Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).

Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian).

E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, V. S. Titov, Properties of central symmetry for diagonal Latin squares, High-performance computing systems and technologies, No. 1 (8), 2018, pp. 74-78. (in Russian)

E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, V. S. Titov, Central Symmetry Properties for Diagonal Latin Squares, Problems of Information Technology, No. 2, 2019, pp. 3-8. doi: 10.25045/jpit.v10.i2.01.

Index entries for sequences related to Latin squares and rectangles

FORMULA

a(n) = A293778(n) / n!.

EXAMPLE

0 1 2 3 4 5 6 7 8

6 3 0 2 7 8 1 4 5

3 2 1 8 6 7 0 5 4

7 8 6 5 1 3 4 0 2

8 6 4 7 2 0 5 3 1

2 7 5 6 8 4 3 1 0

5 4 7 0 3 1 8 2 6

4 5 8 1 0 2 7 6 3

1 0 3 4 5 6 2 8 7

CROSSREFS

Cf. A287649, A287650, A293778, A340545.

Sequence in context: A160636 A282626 A206712 * A200704 A257955 A024544

Adjacent sequences:  A293774 A293775 A293776 * A293778 A293779 A293780

KEYWORD

nonn,more,hard

AUTHOR

Eduard I. Vatutin, Oct 16 2017

STATUS

approved

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Last modified August 11 00:55 EDT 2022. Contains 356046 sequences. (Running on oeis4.)