A287764 -id:A287764 - cp's OEIS frontend

A340550 Number of main classes of diagonal Latin squares of order n that contain a doubly symmetric square.

1, 0, 0, 1, 0, 0, 0, 47, 0, 0, 0

Offset: 1

Eduard I. Vatutin, Jan 11 2021

A doubly symmetric square has symmetries in both the horizontal and vertical planes (see A292517).

Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(n) <= A340545(n). - Eduard I. Vatutin, May 28 2021

			An example of a doubly symmetric diagonal Latin square:
  0 1 2 3 4 5 6 7
  3 2 7 6 1 0 5 4
  2 3 1 0 7 6 4 5
  6 7 5 4 3 2 0 1
  7 6 3 2 5 4 1 0
  4 5 0 1 6 7 2 3
  5 4 6 7 0 1 3 2
  1 0 4 5 2 3 7 6
In the horizontal direction there is a one-to-one correspondence between elements 0 and 7, 1 and 6, 2 and 5, 3 and 4.
In the vertical direction there is also a correspondence between elements 0 and 1, 2 and 4, 6 and 7, 3 and 5.

Eduard I. Vatutin, On the number of main classes of one plane and double plane symmetric diagonal Latin squares of orders 1-11 (in Russian).
Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (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)
Index entries for sequences related to Latin squares and rectangles.

Cf. A287764, A292517, A287650, A340545, A340546.

Name clarified by Andrew Howroyd, Oct 22 2023

A299785 Minimum size of a main class for diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 48, 480, 23040, 161280, 3870720

Offset: 1

Eduard I. Vatutin, Jan 21 2019

0 <= a(n) <= A299787(n). - Eduard I. Vatutin, Jun 08 2020
a(9) <= 17418240; a(10) <= 27869184000. - Eduard I. Vatutin, Oct 05 2020
a(11) <= 61312204800, a(12) <= 22072393728000, a(13) <= 47823519744000. - Eduard I. Vatutin, May 31 2021

			From _Eduard I. Vatutin_, Oct 05 2020: (Start)
The following DLS of order 9 has a main class with cardinality 48*9! = 17418240:
  0 1 2 3 4 5 6 7 8
  2 4 3 0 7 6 8 1 5
  6 2 8 5 3 4 7 0 1
  4 6 7 1 8 2 3 5 0
  1 5 4 7 6 0 2 8 3
  7 8 1 4 5 3 0 6 2
  3 7 0 2 1 8 5 4 6
  8 3 5 6 0 7 1 2 4
  5 0 6 8 2 1 4 3 7
The following DLS of order 10 has a main class with cardinality 7680*10! = 27869184000:
  0 1 2 3 4 5 6 7 8 9
  1 2 0 4 3 6 5 9 7 8
  2 0 3 5 8 1 4 6 9 7
  4 6 9 7 1 8 2 0 3 5
  9 7 8 6 5 4 3 1 2 0
  3 4 7 8 0 9 1 2 5 6
  6 9 4 1 7 2 8 5 0 3
  7 8 5 0 6 3 9 4 1 2
  5 3 1 9 2 7 0 8 6 4
  8 5 6 2 9 0 7 3 4 1
(End)

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
E. I. Vatutin, About the upper bound of the minimal size of main class for diagonal Latin squares of order 9 (in Russian).
E. I. Vatutin, About the upper bound of the minimal size of main class for diagonal Latin squares of order 10 (in Russian).
E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Supercomputing Days Russia 2018, Moscow, Moscow State University, 2018, pp. 933-942.
E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586.
Eduard I. Vatutin, About the relationship between the minimal and maximal cardinality of main classes for diagonal Latin squares (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A287764, A299783, A299784, A299787.

a(n) = A299783(n) * n!.
From Eduard I. Vatutin, May 31 2021: (Start)
a(n) = A299787(n) for 1 <= n <= 5.
a(6) = A299787(6)/3.
a(7) = A299787(7)/6.
a(8) = A299787(8)/16.
a(9) = A299787(9)/32.
a(10) = A299787(10)/2.
a(11) = A299787(11)/10.
a(12) = A299787(12)/4.
a(13) = A299787(13)/24. (End)

A340545 Number of main classes of centrally symmetric diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 32, 301, 430090

Offset: 1

Eduard I. Vatutin, Jan 11 2021

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 numbering of rows and columns from 0 to n-1).
It seems that a(n)=0 for n==2 (mod 4).
Centrally symmetric Latin squares are Latin squares, so a(n) <= A287764(n).
The canonical form (CF) of a square is the lexicographically minimal item within the corresponding main class of diagonal Latin square.
Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that A340550(n) <= a(n). - Eduard I. Vatutin, May 28 2021

			For n=4 there is a single CF:
  0 1 2 3
  2 3 0 1
  3 2 1 0
  1 0 3 2
so a(4)=1.
For n=5 there are two different CFs:
  0 1 2 3 4   0 1 2 3 4
  2 3 4 0 1   1 3 4 2 0
  4 0 1 2 3   4 2 1 0 3
  1 2 3 4 0   2 0 3 4 1
  3 4 0 1 2   3 4 0 1 2
so a(5)=2.
Example of a centrally symmetric diagonal Latin square of order n=9:
  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

E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, and 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, and 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.
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)
E. I. Vatutin, About the number of main classes of centrally symmetric diagonal Latin squares of orders 1-9 (in Russian).
Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A293777, A293778, A287764, A340550.

A345761 a(n) is the number of distinct numbers of orthogonal diagonal mates that a diagonal Latin squares of order n can have.

Original entry on oeis.org

1, 0, 0, 1, 2, 1, 3, 31, 99

Offset: 1

Eduard I. Vatutin, Jun 26 2021

a(n) <= A287695(n) + 1.
a(n) <= A287764(n).
a(10) >= 10. It seems that a(10) = 10 due to long computational experiments within the Gerasim@Home volunteer distributed computing project did not reveal the existence of diagonal Latin squares of order 10 with the number of orthogonal diagonal Latin squares different from {0, 1, 2, 3, 4, 5, 6, 7, 8, 10}.
a(11) >= 112, a(12) >= 5079. - Eduard I. Vatutin, Nov 02 2021, updated Jan 23 2023

			For n=7 the number of orthogonal diagonal Latin squares that a diagonal Latin square of order 7 may have is 0, 1, or 3. Since there are 3 distinct values, a(7)=3.

Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of orders 1-7 (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of order 8 (in Russian).
Eduard I. Vatutin, About the spectrum of orthogonal diagonal Latin squares for one diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, About the lower bound for a spectrum of orthogonal diagonal Latin squares for one diagonal Latin squares of order 10 (in Russian).
Eduard I. Vatutin, About the lower bound for a spectrum of orthogonal diagonal Latin squares for one diagonal Latin squares of order 11 (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian)
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9 (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022. pp. 314-315.
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, I. I. Kurochkin, Heuristic method for getting approximations of spectra of numerical characteristics for diagonal Latin squares, Intellectual information systems: trends, problems, prospects, Kursk, 2022. pp. 35-41. (in Russian)
Eduard I. Vatutin, Proving lists (1, 4, 5, 6, 7, 8, 9, 10, 11, 12).
Index entries for sequences related to Latin squares and rectangles.

Cf. A287695, A287764, A309344, A344105, A345370, A345760.

A343866 Number of inequivalent cyclic diagonal Latin squares of order 2n+1 up to rotations, reflections and permutation of symbols.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 3, 0, 4, 4, 0, 5, 3, 0, 7, 7, 0, 2, 9, 0, 10, 10, 0, 11, 7, 0, 13, 4, 0, 14, 15, 0, 6, 16, 0, 17, 18, 0, 8, 19, 0, 20, 8, 0, 22, 10, 0, 8, 24, 0, 25, 25, 0, 26, 27, 0, 28, 10, 0, 14, 22, 0, 13, 31, 0, 32, 16, 0, 34, 34, 0, 20, 14, 0, 37, 37, 0, 14, 39, 0, 20

Offset: 0

Andrew Howroyd, May 02 2021

nonn

Also the number of main classes of diagonal Latin squares of order 2n+1 that contain a cyclic Latin square. Compare A341585.

			a(12) = 3 since there are A123565(25) = 10 cyclic diagonal Latin squares whose first row is in ascending order. Each of these is uniquely defined by the step between rows and form 5 pairs by horizontal or vertical reflection (negating the step between rows). Up to exchanging rows with columns there are 3 distinct classes, so a(12) = 3.

Cf. A123565, A287764, A338562, A341585.

iscanon(n,k,g) = k <= vecmin(g*k%n) && k <= vecmin(g*lift(1/Mod(k,n))%n)
a(n)={if(n==0, 1, my(m=2*n+1); sum(k=1, m-1, gcd(m,k)==1 && gcd(m,k-1)==1 && gcd(m,k+1)==1 && iscanon(m, k, [1,-1])))}

a((p-1)/2) = A341585((p-1)/2) for odd prime p.

A366331 Number of main classes of diagonal Latin squares of order 2n+1 that contain a horizontally semicyclic Latin square.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 20, 0, 272, 1208, 0, 127334, 1958084, 0

Offset: 0

Eduard I. Vatutin, Oct 07 2023

A 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).

			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)

Eduard I. Vatutin, About the horizontally and vertically semicyclic diagonal Latin squares enumeration (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of different types of cyclic diagonal Latin squares (in Russian).
Eduard I. Vatutin, About the number of main classes of semicyclic diagonal Latin squares of order 17 (in Russian).
Eduard I. Vatutin, About the number of main classes of semicyclic diagonal Latin squares of order 19 (in Russian).
Eduard I. Vatutin, Lists of canonical forms of semicyclic diagonal Latin squares of orders 5-19.
Index entries for sequences related to Latin squares and rectangles.

Cf. A071607, A123565, A287764, A338562, A342990, A343866.

a(11)-a(13) from Andrew Howroyd, Nov 02 2023

A337309 Number of main classes of bachelor diagonal Latin squares of order n.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 967, 4871991, 3292326080087

Offset: 1

Eduard I. Vatutin, Aug 22 2020

A bachelor diagonal Latin square is one with no orthogonal mate.

Eduard I. Vatutin, About the number of main classes of bachelor diagonal Latin squares of orders 1-8 (in Russian).
Eduard I. Vatutin, Natalia N. Nikitina, Maxim O. Manzuk, First results of an experiment on studying the properties of DLS of order 9 in the volunteer distributed computing projects Gerasim@Home and RakeSearch (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)
Index entries for sequences related to Latin squares and rectangles.

Cf. A305568, A305569, A330391, A287764.

a(n) = A287764(n) - A330391(n).

a(9) added by Eduard I. Vatutin, Dec 08 2020

cp's OEIS Frontend

A340550 Number of main classes of diagonal Latin squares of order n that contain a doubly symmetric square.

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A299785 Minimum size of a main class for diagonal Latin squares of order n.

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A340545 Number of main classes of centrally symmetric diagonal Latin squares of order n.

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A345761 a(n) is the number of distinct numbers of orthogonal diagonal mates that a diagonal Latin squares of order n can have.

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A343866 Number of inequivalent cyclic diagonal Latin squares of order 2n+1 up to rotations, reflections and permutation of symbols.

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A366331 Number of main classes of diagonal Latin squares of order 2n+1 that contain a horizontally semicyclic Latin square.

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A337309 Number of main classes of bachelor diagonal Latin squares of order n.

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