A338620 -id:A338620 - cp's OEIS frontend

A071607 Number of strong complete mappings of the cyclic group Z_{2n+1}.

1, 0, 2, 4, 0, 8, 348, 0, 8276, 43184, 0, 5602176, 78309000, 0, 20893691564, 432417667152, 0

Offset: 0

J. Hsiang, D. F. Hsu and Y. P. Shieh (arping(AT)turing.csie.ntu.edu.tw), Jun 03 2002

A strong complete mapping of a cyclic group (Z_n,+) is a permutation f(x) of Z_n such that f(0)=0 and that f(x)-x and f(x)+x are both permutations.
a(n) is the number of solutions of the toroidal n-queen problem (A007705) with 2n+1 queens and one queen in the top-left corner.
Also a(n) is the number of horizontally or vertically semicyclic diagonal Latin squares of order 2n+1 with the first row in ascending order. 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 A123565) 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. - Eduard I. Vatutin, Jan 25 2022

			f(x)=2x in (Z_7,+) is a strong complete mapping of Z_7 since f(0)=0 and both f(x)-x (=x) and f(x)+x (=3x) are permutations of Z_7.
From _Eduard I. Vatutin_, Jan 25 2022: (Start)
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=1)
   3  4  5  6  7  8  9 10 11 12  0  1  2  (d=3)
   5  6  7  8  9 10 11 12  0  1  2  3  4  (d=5)
   7  8  9 10 11 12  0  1  2  3  4  5  6  (d=7)
   9 10 11 12  0  1  2  3  4  5  6  7  8  (d=9)
  11 12  0  1  2  3  4  5  6  7  8  9 10  (d=11)
.
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)
(End)
From _Eduard I. Vatutin_, Apr 09 2024: (Start)
Example of N-queens problem on toroidal board, N=2*2+1=5, a(2)=2, given by knight with (+1,+2) and (+1,+3) movement parameters starting from top left corner:
.
+-----------+ +-----------+
| Q . . . . | | Q . . . . |
| . . Q . . | | . . . Q . |
| . . . . Q | | . Q . . . |
| . Q . . . | | . . . . Q |
| . . . Q . | | . . Q . . |
+-----------+ +-----------+
.
Example of N-queens problem on toroidal board, N=2*3+1=7, a(3)=4, given by knight with (+1,+2), (+1,+3), (+1,+4), (+1,+5) movement parameters starting from top left corner:
.
+---------------+ +---------------+ +---------------+ +---------------+
| Q . . . . . . | | Q . . . . . . | | Q . . . . . . | | Q . . . . . . |
| . . Q . . . . | | . . . Q . . . | | . . . . Q . . | | . . . . . Q . |
| . . . . Q . . | | . . . . . . Q | | . Q . . . . . | | . . . Q . . . |
| . . . . . . Q | | . . Q . . . . | | . . . . . Q . | | . Q . . . . . |
| . Q . . . . . | | . . . . . Q . | | . . Q . . . . | | . . . . . . Q |
| . . . Q . . . | | . Q . . . . . | | . . . . . . Q | | . . . . Q . . |
| . . . . . Q . | | . . . . Q . . | | . . . Q . . . | | . . Q . . . . |
+---------------+ +---------------+ +---------------+ +---------------+
(End)

Anthony B. Evans,"Orthomorphism Graphs of Groups", vol. 1535 of Lecture Notes in Mathematics, Springer-Verlag, 1991.
Y. P. Shieh, J. Hsiang and D. F. Hsu, "On the enumeration of Abelian k-complete mappings", vol. 144 of Congressus Numerantium, 2000, pp. 67-88.

Jieh Hsiang, Yuh Pyng Shieh, and Yao Chiang Chen, Cyclic complete mappings counting problems, in PaPS: Problems and Problem Sets for ATP Workshop in conjunction with CADE-18 and FLoC 2002.
Jieh Hsiang, YuhPyng Shieh, and YaoChiang Chen, Cyclic Complete Mappings Counting Problems, National Taiwan University 2014/8/21.
D. Novakovic, Computation of the number of complete mappings for permutations, Cybernetics & System Analysis, No. 2, v. 36 (2000), pp. 244-247.
I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), pp. 629-639.
Eduard I. Vatutin, About the horizontally and vertically semicyclic diagonal Latin squares enumeration (in Russian).
Eduard 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)
Index entries for sequences related to Latin squares and rectangles.

Cf. A007705, A051906, A123565, A338620, A342990, A343867.

a(n) = A007705(n) / (2*n+1).
a(n) = A342990(n) / (2*n+1)!. - Eduard I. Vatutin, Mar 10 2022
a(n) = A051906(2*n+1) / (2*n+1). - Eduard I. Vatutin, Apr 09 2024

a(15)-a(16) added using A007705 by Andrew Howroyd, May 07 2021

A342306 Number of pandiagonal Latin squares of order 2n+1.

Original entry on oeis.org

1, 0, 240, 20160, 0, 319334400, 77127879628800, 0

Offset: 0

Eduard I. Vatutin, Mar 08 2021

A pandiagonal Latin square is a Latin square in which the diagonal, antidiagonal and all broken diagonals and antidiagonals are transversals.
For orders 5, 7 and 11 all pandiagonal Latin squares are cyclic, so a(n) = A338562(n) for n < 6. For n=6 (order 13) this is not true (from Dabbaghian and Wu).
Pandiagonal Latin squares exist only for odd orders not divisible by 3. - Andrew Howroyd, May 26 2021

			Example of a cyclic pandiagonal Latin square of order 5:
  0 1 2 3 4
  2 3 4 0 1
  4 0 1 2 3
  1 2 3 4 0
  3 4 0 1 2
Example of a cyclic pandiagonal Latin square of order 7:
  0 1 2 3 4 5 6
  2 3 4 5 6 0 1
  4 5 6 0 1 2 3
  6 0 1 2 3 4 5
  1 2 3 4 5 6 0
  3 4 5 6 0 1 2
  5 6 0 1 2 3 4
Example of a cyclic pandiagonal Latin square of order 11:
   0  1  2  3  4  5  6  7  8  9 10
   2  3  4  5  6  7  8  9 10  0  1
   4  5  6  7  8  9 10  0  1  2  3
   6  7  8  9 10  0  1  2  3  4  5
   8  9 10  0  1  2  3  4  5  6  7
  10  0  1  2  3  4  5  6  7  8  9
   1  2  3  4  5  6  7  8  9 10  0
   3  4  5  6  7  8  9 10  0  1  2
   5  6  7  8  9 10  0  1  2  3  4
   7  8  9 10  0  1  2  3  4  5  6
   9 10  0  1  2  3  4  5  6  7  8
For order 13 there is a square
   7  1  0  3  6  5 12  2  8  9 10 11  4
   2  3  4 10  0  7  6  9 12 11  5  8  1
   4 11  1  7  8  9 10  3  6  0 12  2  5
   6  5  8 11 10  4  7  0  1  2  3  9 12
   8  9  2  5 12 11  1  4  3 10  0  6  7
   3  6 12  0  1  2  8 11  5  4  7 10  9
  10  0  3  2  9 12  5  6  7  8  1  4 11
   1  7 10  4  3  6  9  8  2  5 11 12  0
  11  4  5  6  7  0  3 10  9 12  2  1  8
   5  8  7  1  4 10 11 12  0  6  9  3  2
  12  2  9  8 11  1  0  7 10  3  4  5  6
   9 10 11 12  5  8  2  1  4  7  6  0  3
   0 12  6  9  2  3  4  5 11  1  8  7 10
that is pandiagonal but not cyclic (Dabbaghian and Wu).

A.O.L. Atkin, L. Hay, and R. G. Larson, Enumeration and construction of pandiagonal Latin squares of prime order, Computers & Mathematics with Applications, Volume. 9, Iss. 2, 1983, pp. 267-292.
Vahid Dabbaghian and Tiankuang Wu, Constructing non-cyclic pandiagonal Latin squares of prime orders, Journal of Discrete Algorithms 30, 2015.

Cf. A338562, A338620.

a(n) = A338620(n) * (2*n+1)!.

Zero terms for even orders removed by Andrew Howroyd, May 26 2021

A343867 Number of semicyclic pandiagonal Latin squares of order 2*n+1 with the first row in ascending order.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1560, 0, 34000, 175104, 0, 22417824, 313235960, 0, 83574857328, 1729671003296

Offset: 0

Andrew Howroyd, May 08 2021

Pandiagonal Latin squares exist only for odd orders not divisible by 3. All pandiagonal Latin squares for orders less than 13 are cyclic which are not counted by this sequence.
Semicyclic Latin squares are defined in the Atkin reference where the first nonzero term of this sequence is given. They are cyclic in a single direction. The direction can be horizontal or vertical or any other step such as a knights move.
Each symbol in a semicyclic Latin square occupies the same pattern of squares up to translation on the torus which in the case of a pandiagonal square is a solution to the toroidal n-queens problem.
For prime 2n+1, a(n) is a multiple of 2n+1.

			The following is an example of an order 13 semicyclic square with a step of (1,4). This means moving down one row and across by 4 columns increases the cell value by 1 modulo 13. Symbols can be relabeled to give a square with the first row in ascending order.
   0 11  1  7  5  9  3 10  4  8  6 12  2
   9  7  0  3  1 12  2  8  6 10  4 11  5
  11  5 12  6 10  8  1  4  2  0  3  9  7
   1  4 10  8 12  6  0  7 11  9  2  5  3
  10  3  6  4  2  5 11  9  0  7  1  8 12
   8  2  9  0 11  4  7  5  3  6 12 10  1
   7  0 11  2  9  3 10  1 12  5  8  6  4
   6  9  7  5  8  1 12  3 10  4 11  2  0
   5 12  3  1  7 10  8  6  9  2  0  4 11
   3  1  5 12  6  0  4  2  8 11  9  7 10
  12 10  8 11  4  2  6  0  7  1  5  3  9
   2  6  4 10  0 11  9 12  5  3  7  1  8
   4  8  2  9  3  7  5 11  1 12 10  0  6
...
a(12) = 4*(A071607(12) - A123565(25)) + 11240. - _Jim White_, Jul 22 2021
a(14) = 4*(A071607(14) - A123565(29)) + 91176. - _Jim White_, Jul 24 2021
a(15) = 4*(A071607(15) - A123565(31)) + 334800. - _Jim White_, Aug 03 2021

A. O. L. Atkin, L. Hay, and R. G. Larson, Enumeration and construction of pandiagonal Latin squares of prime order, Computers & Mathematics with Applications, Volume. 9, Iss. 2, 1983, pp. 267-292.
Andrew Howroyd, PARI Program for Initial Terms.
Natalia Makarova from Harry White, 1560 semi-cyclic Latin squares of order 13.
Natalia Makarova from Harry White, 34000 semi-cyclic Latin squares of order 17.
Eduard I. Vatutin, 175104 semi-cyclic Latin squares of order 19.

Cf. A071607, A123565 (cyclic), A338620, A343868.

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a(n) >= 4*(A071607(n) - A123565(2*n+1)).

a(12)-a(15) from Jim White, Aug 03 2021

cp's OEIS Frontend

A071607 Number of strong complete mappings of the cyclic group Z_{2n+1}.

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A342306 Number of pandiagonal Latin squares of order 2n+1.

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A343867 Number of semicyclic pandiagonal Latin squares of order 2*n+1 with the first row in ascending order.

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