A342990 Number of horizontally or vertically semicyclic diagonal Latin squares of order 2n+1.

1, 0, 240, 20160, 0, 319334400, 2167003238400, 0, 2943669154922496000, 5253122016055001088000, 0, 144827547726179682893168640000, 1214667347283206181421056000000000, 0, 184737047979495031539522261089255424000000, 3555700708206908663181998415125686517760000000, 0

Offset: 0

Author

Eduard I. Vatutin, Jan 27 2022

Keywords

• nonn
• more
• hard

Comments

Horizontally semicyclic diagonal Latin square is a square where each row r(i) is a cyclic shift of the first row r(0) by some value d(i) (see example). Vertically semicyclic diagonal Latin square is a square where each column c(i) is a cyclic shift of the first column c(0) by some value d(i). Cyclic diagonal Latin squares (see A338562) fall under the definition of vertically and horizontally semicyclic diagonal Latin squares simultaneously, in this type of squares each row r(i) is obtained from the previous one r(i-1) using cyclic shift by some value d. Definition from A343867 includes this type of squares but not only it.

Examples

			Example of cyclic diagonal Latin square of order 13:
   0  1  2  3  4  5  6  7  8  9 10 11 12
   2  3  4  5  6  7  8  9 10 11 12  0  1  (d=2)
   4  5  6  7  8  9 10 11 12  0  1  2  3  (d=4)
   6  7  8  9 10 11 12  0  1  2  3  4  5  (d=6)
   8  9 10 11 12  0  1  2  3  4  5  6  7  (d=8)
  10 11 12  0  1  2  3  4  5  6  7  8  9  (d=10)
  12  0  1  2  3  4  5  6  7  8  9 10 11  (d=12)
   1  2  3  4  5  6  7  8  9 10 11 12  0  (d=14 ==  1 (mod 13))
   3  4  5  6  7  8  9 10 11 12  0  1  2  (d=16 ==  3 (mod 13))
   5  6  7  8  9 10 11 12  0  1  2  3  4  (d=18 ==  5 (mod 13))
   7  8  9 10 11 12  0  1  2  3  4  5  6  (d=20 ==  7 (mod 13))
   9 10 11 12  0  1  2  3  4  5  6  7  8  (d=22 ==  9 (mod 13))
  11 12  0  1  2  3  4  5  6  7  8  9 10  (d=24 == 11 (mod 13))
Example of horizontally semicyclic diagonal Latin square of order 13:
   0  1  2  3  4  5  6  7  8  9 10 11 12
   2  3  4  5  6  7  8  9 10 11 12  0  1  (d=2)
   4  5  6  7  8  9 10 11 12  0  1  2  3  (d=4)
   9 10 11 12  0  1  2  3  4  5  6  7  8  (d=9)
   7  8  9 10 11 12  0  1  2  3  4  5  6  (d=7)
  12  0  1  2  3  4  5  6  7  8  9 10 11  (d=12)
   3  4  5  6  7  8  9 10 11 12  0  1  2  (d=3)
  11 12  0  1  2  3  4  5  6  7  8  9 10  (d=11)
   6  7  8  9 10 11 12  0  1  2  3  4  5  (d=6)
   1  2  3  4  5  6  7  8  9 10 11 12  0  (d=1)
   5  6  7  8  9 10 11 12  0  1  2  3  4  (d=5)
  10 11 12  0  1  2  3  4  5  6  7  8  9  (d=10)
   8  9 10 11 12  0  1  2  3  4  5  6  7  (d=8)
		

Links

• Eduard I. Vatutin, About the horizontally and vertically semicyclic diagonal Latin squares enumeration (in Russian).
• E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
• Eduard I. Vatutin, Numerical formula between number of horizontally or vertically semicyclic diagonal Latin squares and number of toroidal n-queens problem solutions (in Russian).
• Index entries for sequences related to Latin squares and rectangles.

Crossrefs

Cf. A007705, A071607, A338562, A343867.

Formula

a(n) = A071607(n) * (2*n+1)!.

a(n) = A007705(n) * (2n)!. - Eduard I. Vatutin, Mar 15 2024