A274806 -id:A274806 - cp's OEIS frontend

A274171 Number of diagonal Latin squares of order n with the first row in order.

1, 0, 0, 2, 8, 128, 171200, 7447587840, 5056994653507584

Offset: 1

Eduard I. Vatutin, Jul 07 2016

A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element. - Andrew Howroyd, Sep 29 2020

			The a(4) = 2 diagonal Latin squares are:
   0 1 2 3   0 1 2 3
   2 3 0 1   3 2 1 0
   3 2 1 0   1 0 3 2
   1 0 3 2   2 3 0 1
.
The a(5) = 8 diagonal Latin squares are:
   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4
   1 3 4 2 0   1 4 3 0 2   2 3 4 0 1   2 4 1 0 3
   4 2 1 0 3   3 2 1 4 0   4 0 1 2 3   4 0 3 2 1
   2 0 3 4 1   4 3 0 2 1   1 2 3 4 0   3 2 4 1 0
   3 4 0 1 2   2 0 4 1 3   3 4 0 1 2   1 3 0 4 2
.
   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4
   3 4 0 1 2   3 4 1 2 0   4 2 0 1 3   4 2 3 0 1
   1 2 3 4 0   4 2 3 0 1   1 4 3 2 0   3 4 1 2 0
   4 0 1 2 3   2 0 4 1 3   3 0 1 4 2   1 3 0 4 2
   2 3 4 0 1   1 3 0 4 2   2 3 4 0 1   2 0 4 1 3

S. E. Kochemazov, E. I. Vatutin, and O. S. Zaikin, Fast Algorithm for Enumerating Diagonal Latin Squares of Small Order, arXiv:1709.02599 [math.CO], 2017.
S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
M. O. Manzuk and N. N. Nikitina, About the number of diagonal Latin squares of order 9 as a one of results of RakeSearch distributed computing project
Eduard I. Vatutin, a(9) value fixed after
E. I. Vatutin, Enumerating the diagonal Latin squares of order 8 using equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian)
E. I. Vatutin, Enumerating the diagonal Latin squares of order 9 using Gerasim@Home volunteer distributed computing project, equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian)
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).
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Applying Volunteer and Parallel Computing for Enumerating Diagonal Latin Squares of Order 9, Parallel Computational Technologies. PCT 2017. Communications in Computer and Information Science, vol. 753, pp. 114-129. doi: 10.1007/978-3-319-67035-5_9.
Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S.Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, and Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzuk, S. E. Kochemazov and V. S. Titov, Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares, Proceedings of Distributed Computing and grid-technologies in science and education (GRID'16), JINR, Dubna, 2016, pp. 114-115.
Vatutin E. I., Zaikin O. S., Zhuravlev A. D., Manzuk M. O., Kochemazov S. E., and Titov V. S., The effect of filling cells order to the rate of generation of diagonal Latin squares, Information-measuring and diagnosing control systems (Diagnostics - 2016). Kursk: SWSU, 2016. pp. 33-39 (in Russian).
E. I. Vatutin, V. S. Titov, O. S. Zaikin, S. E. Kochemazov, S. U. Valyaev, A. D. Zhuravlev, and M. O. Manzuk, Using grid systems for enumerating combinatorial objects with example of diagonal Latin squares, Information technologies and mathematical modeling of systems (2016), pp. 154-157, (in Russian).
Vatutin E.I., Zaikin O.S., Zhuravlev A.D., Manzyuk M.O., Kochemazov S.E., and Titov V.S., Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares, CEUR Workshop proceedings. Selected Papers of the 7th International Conference Distributed Computing and Grid-technologies in Science and Education. 2017. Vol. 1787. pp. 486-490. urn:nbn:de:0074-1787-5.
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. A000315, A000479, A274806, A287764, A309283.

a(n) = A274806(n)/n!.

a(9) added from Vatutin et al. (2016) by Max Alekseyev, Oct 05 2016

a(9) corrected by Eduard I. Vatutin, Oct 20 2016

A287648 Maximum number of diagonal transversals in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 4, 5, 6, 27, 120, 333

Offset: 1

Eduard I. Vatutin, May 29 2017

From Eduard I. Vatutin, Oct 04 2020: (Start)
A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element.
A diagonal transversal is a transversal that includes exactly one element from the main diagonal and exactly one from the antidiagonal. For squares of odd orders, these elements can coincide at the intersection of the diagonals. (End)
A007016 is an upper bound for the number of diagonal transversals in a Latin square: A287647(n) <= a(n) <= A007016(n). - Eduard I. Vatutin, Jan 02 2020
a(11) >= 4828, a(12) >= 24901, a(13) >= 131106, a(14) >= 364596, a(15) >= 389318. - Natalia Makarova, Tomáš Brada, Harry White, Oct 04 2020
a(16) >= 32172800, a(18) >= 280308432. - Natalia Makarova, Tomáš Brada, Dec 25 2020
a(12) >= 28496. - Natalia Makarova, Harry White, Jan 23 2021
a(14) >= 380718, a(20) >= 90010806304, a(21) >= 51162162017, a(22) >= 3227747329246. The number of D-transversals for orders 20 - 22 was calculated by a volunteer. - Natalia Makarova, Tomáš Brada, Harry White, Mar 17 2021
All cyclic diagonal Latin squares (see A338562) are diagonal Latin squares, so A342997((n-1)/2) <= a(n). - Eduard I. Vatutin, Apr 26 2021
a(14) >= 383578, a(15) >= 398974. - Natalia Makarova, Tomáš Brada, Jan 13 2022
a(10) >= 890, a(12) >= 30192, a(14) >= 490218, a(15) >= 4620434, a(17) >= 204995269, a(18) >= 281593874, a(19) >= 11254190082. - Eduard I. Vatutin, Jul 22 2020, updated Mar 01 2025
For most orders n, at least one diagonal Latin square with the maximal number of diagonal transversals has an orthogonal mate and a(n) = A360220(n). Known exceptions: n=6 and n=10. - Eduard I. Vatutin, Feb 17 2023

			For example, the diagonal Latin square
  0 1 2 3
  3 2 1 0
  1 0 3 2
  2 3 0 1
has 4 diagonal transversals:
  0 . . .    . 1 . .    . . 2 .    . . . 3
  . . 1 .    . . . 0    3 . . .    . 2 . .
  . . . 2    . . 3 .    . 0 . .    1 . . .
  . 3 . .    2 . . .    . . . 1    . . 0 .
In addition there are 4 other transversals that are not diagonal transversals and are therefore not included here.
From _Natalia Makarova_, Oct 04 2020: (Start)
The following DLS of order 14 has 364596 diagonal transversals:
   0  7  6 11  9  3  4  5  2 12 13  8 10  1
   6  1 11  5 10 12  2  3  9  7  4 13  0  8
   5 11  2 12  8  1  7 10  0  6  9  3 13  4
  13  6  5  3  1 10  9 12  7  0  2  4  8 11
  12  3 10  1  4 13  8  6 11  5  0  7  2  9
  10 12  1  8  2  5 11 13  4  3  6  0  9  7
   9  2  7  0  5 11  6  8 13  4  1 10  3 12
   4 13  3  9  6  0 10  7  1  8 12  2 11  5
   2  4  9 10 11  6  1  0  8 13  7 12  5  3
   1 10  8 13 12  2  5  4  3  9 11  6  7  0
   3  5 12  7 13  8  0  1  6 11 10  9  4  2
   8  0 13  4  7  9  3  2 12 10  5 11  1  6
   7  9  0  6  3  4 13 11  5  2  8  1 12 10
  11  8  4  2  0  7 12  9 10  1  3  5  6 13
(End)

J. W. Brown, F. Cherry, L. Most, M. Most, E. T. Parker, and W. D. Wallis, Completion of the spectrum of orthogonal diagonal Latin squares, Lecture notes in pure and applied mathematics. 1992. Vol. 139. pp. 43-49.

Tomáš Brada, Top 10 CF-ODLK with most orthogonal mates
Natalia Makarova, Most perfect diagonal Latin square of order 9 with 333 diagonal transversals
Natalia Makarova, ODLS of order n>10
Natalia Makarova, DLS with maximum of D-transversals
Natalia Makarova, DLS of orders n = 11 - 22 with known maximum of D-transversals
Natalia Makarova, Spectrum by D-transversals for the 14th order DLS
Natalia Makarova, Spectrum by D-transversals for the 15th order DLS
E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
Eduard I. Vatutin, Enumerating the Main Classes of Cyclic and Pandiagonal Latin Squares, Recognition - 2021, pp. 77-79. (in Russian)
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, Maxim Manzuk, Alexander Albertian, Ilya Kurochkin, Alexander Kripachev, and Alexey Pykhtin, Diagonalization and Canonization of Latin Squares, Supercomputing, Russian Supercomputing Days (RuSCDays 2023) Rev. Selected Papers Part II, LCNS Vol. 14389, Springer, Cham, 48-61.
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Estimating of combinatorial characteristics for diagonal Latin squares, Recognition — 2017 (2017), pp. 98-100 (in Russian).
Eduard I. Vatutin, Stepan E. Kochemazov, Oleg S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, and Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and S. Yu. Valyaev, Enumerating the Transversals for Diagonal Latin Squares of Small Order, CEUR Workshop Proceedings. Proceedings of the Third International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017. pp. 6-14. urn:nbn:de:0074-1973-0.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and S. Yu. Valyaev, Using Volunteer Computing to Study Some Features of Diagonal Latin Squares, Open Engineering. Vol. 7. Iss. 1. 2017. pp. 453-460. DOI: 10.1515/eng-2017-0052.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev, and V. S. Titov, Estimating the Number of Transversals for Diagonal Latin Squares of Small Order, Telecommunications. 2018. No. 1. pp. 12-21 (in Russian).
E. I. Vatutin, About the upper bound of number of diagonal transversals for diagonal Latin squares of order 10 (in Russian).
E. I. Vatutin, About the upper bound of number of diagonal transversals for diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, Natalia N. Nikitina, and 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, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, 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.
Eduard I. Vatutin, Best known examples.
Index entries for sequences related to Latin squares and rectangles.

Cf. A274806, A287644, A287645, A287647, A342997, A345370.

a(8) added by Eduard I. Vatutin, Oct 29 2017

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

A305571 Number of diagonal Latin squares of order n with at least one orthogonal diagonal mate.

Original entry on oeis.org

1, 0, 0, 48, 480, 0, 1290240, 25484820480, 34482663290880

Offset: 1

Eduard I. Vatutin, Jun 05 2018

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (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)
E. Vatutin and A. Belyshev, Enumerating the Orthogonal Diagonal Latin Squares of Small Order for Different Types of Orthogonality, Communications in Computer and Information Science, Vol. 1331, Springer, 2020, pp. 586-597.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and I. I. Citerra, Estimation of the probability of finding orthogonal diagonal Latin squares among general diagonal Latin squares, Recognition - 2018. Kursk: SWSU, 2018. pp. 72-74. (in Russian).
Eduard I. Vatutin, Natalia N. Nikitina, and 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).
Eduard I. Vatutin, Natalia N. Nikitina, and Maxim O. Manzuk, Additional calculated 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, List of all main classes of orthogonal diagonal Latin squares of orders 1-8.
Index entries for sequences related to Latin squares and rectangles.

Cf. A274806, A305569, A305570.

a(n) = A305570(n) * n!.

a(n) = A274806(n) - A305569(n).

a(9) added by Eduard I. Vatutin, Dec 22 2020

A309283 Number of equivalence classes of X-based filling of diagonals in a diagonal Latin square of order n.

Original entry on oeis.org

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

Offset: 0

Eduard I. Vatutin, Jul 06 2020

Used for getting strong canonical forms (SCFs) of the diagonal Latin squares and for fast enumerating of the diagonal Latin squares based on equivalence classes.
K1 = |C[1]|*f[1] + |C[2]|*f[2] + ... + |C[m]|*f[m],
K2 = K1 * n!,
where m = a(n), number of equivalence classes for X-based filling of diagonals in a diagonal Latin square of order n;
C[i], corresponding equivalence classes with cardinalities |C[i]|, 1 <= i <= m;
f[i], the number of diagonal Latin squares corresponds to the each item from equivalence class C[i], 1 <= i <= m;
K1 = A274171(n), number of diagonal Latin squares of order n with fixed first row;
K2 = A274806(n), number of diagonal Latin squares of order n.
For all t>0 a(2*t) = a(2*t+1). - Eduard I. Vatutin, Aug 21 2020
a(14) >= 5225, a(15) >= 5225. - Natalia Makarova, Sep 12 2020
The number of solutions in an equivalence class with the main diagonal in ascending order is at most 4*2^r*r! where r = floor(n/2). This maximum is achieved for orders n >= 10. - Andrew Howroyd, Mar 27 2023

			For order n=4 there are a(4)=2 equivalence classes. First of them C[1] includes two X-based fillings of diagonals
   0..1  0..2
   .13.  .10.
   .02.  .32.
   2..3  1..3
and second C[2] also includes two X-based fillings of diagonals
   0..1  0..2
   .10.  .13.
   .32.  .02.
   2..3  1..3
It is easy to see that f[1] = 0 and f[2] = 1, so K1(4) = A274171(4) = 2*0 + 2*1 = 2 and K2(4) = A274806(4) = K1(4) * 4! = 2 * 24 = 48.

S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
Natalia Makarova, All 67 rules for SN DLS of order 11
Natalia Makarova and Harry White, About unique diagonals for SN DLS of order 14 and 15
E. I. Vatutin, About the number of strong normalized lines of diagonal Latin squares of orders 1-10 (in Russian).
E. I. Vatutin, About the number of strong normalized lines of diagonal Latin squares of order 11 (in Russian).
E. I. Vatutin, About the a(2*t)=a(2*t+1) equality (in Russian).
E. I. Vatutin, About the number of equivalence classes of X-based filling of diagonals in a diagonal Latin squares of order 12 (in Russian).
E. I. Vatutin, About the number of equivalence classes of X-based filling of diagonals in a diagonal Latin squares of order 13 (in Russian).
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).
Index entries for sequences related to Latin squares and rectangles.

Cf. A274171, A274806, A337302, A338084.

a(n) = A338084(floor(n/2)).

a(11) added by Eduard I. Vatutin, Aug 21 2020
a(12)-a(13) by Harry White, added by Natalia Makarova, Sep 12 2020
a(0)=1 prepended by Andrew Howroyd, Oct 31 2020

A307166 Minimum number of loops in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 12, 10, 27, 21, 40

Offset: 1

Eduard I. Vatutin, Mar 27 2019

A loop in a Latin square is a sequence of cells v1=L[i1,j1] -> v2=L[i1,j2] -> v1=L[i2,j2] -> ... -> v2=L[im,j1] -> v1=L[i1,j1] of length 2*m that consists of a pair of values {v1, v2}.
For diagonal Latin squares of order 4 all loops are intercalates. - Eduard I. Vatutin, Oct 05 2020
From Eduard I. Vatutin, Oct 26 2020: (Start)
Every intercalate is a partial loop and every partial loop is a loop, so 0 <= A307163(n) <= A307170(n) <= a(n).
0 <= a(n) <= A307167(n).
(End)

			For example, the square
  2 4 3 5 0 1
  1 0 4 3 2 5
  0 2 5 4 1 3
  5 3 0 1 4 2
  4 5 1 2 3 0
  3 1 2 0 5 4
has a loop
  2 4 . . . .
  . . . . . .
  . 2 . 4 . .
  . . . . . .
  4 . . 2 . .
  . . . . . .
consisting of the sequence of cells L[1,1]=2 -> L[1,2]=4 -> L[3,2]=2 -> L[3,4]=4 -> L[5,4]=2 -> L[5,1]=4 -> L[1,1]=2 with length 6.
The total number of loops for this square is 21.

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
E. I. Vatutin, About the minimum and maximum number of loops in a diagonal Latin squares of order 8 (in Russian).
E. I. Vatutin, On the inequalities of the minimum and maximum numerical characteristics of diagonal Latin squares for intercalates, loops and partial loops (in Russian).
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer (2020), 127-146.
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A274806, A307163, A307164, A307167, A307170.

a(8) added by Eduard I. Vatutin, Oct 05 2020

A307167 Maximum number of loops in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 12, 14, 27, 53, 112

Offset: 1

Eduard I. Vatutin, Mar 27 2019

A loop in a Latin square is a sequence of cells v1=L[i1,j1] -> v2=L[i1,j2] -> v1=L[i2,j2] -> ... -> v2=L[im,j1] -> v1=L[i1,j1] of length 2*m that consists of a pair of values {v1, v2}.

For diagonal Latin squares of order 4 all loops are intercalates. - Eduard I. Vatutin, Oct 05 2020

			For example, the square
  2 4 3 5 0 1
  1 0 4 3 2 5
  0 2 5 4 1 3
  5 3 0 1 4 2
  4 5 1 2 3 0
  3 1 2 0 5 4
has a loop
  2 4 . . . .
  . . . . . .
  . 2 . 4 . .
  . . . . . .
  4 . . 2 . .
  . . . . . .
consisting of the sequence of cells L[1,1]=2 -> L[1,2]=4 -> L[3,2]=2 -> L[3,4]=4 -> L[5,4]=2 -> L[5,1]=4 -> L[1,1]=2 with length 6.
The total number of loops for this square is 21.

Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
Eduard I. Vatutin, About the minimum and maximum number of loops in a diagonal Latin squares of order 8 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer (2020), 127-146.
Index entries for sequences related to Latin squares and rectangles.

Cf. A274806, A307166, A307163, A307164.

a(8) added by Eduard I. Vatutin, Oct 05 2020

A307170 Minimum number of partial loops in a diagonal Latin square of order n.

Original entry on oeis.org

0, 0, 0, 12, 0, 21, 0, 24

Offset: 1

Eduard I. Vatutin, Mar 27 2019

A loop in a Latin square is a sequence of cells v1=L[i1,j1] -> v2=L[i1,j2] -> v1=L[i2,j2] -> ... -> v2=L[im,j1] -> v1=L[i1,j1] of length 2*m that consists of a pair of values {v1, v2}. A partial loop is a loop with length < 2*n.
From Eduard I. Vatutin, Oct 20 2020: (Start)
Every intercalate is a partial loop and every partial loop is a loop, so 0 <= A307163(n) <= a(n) <= A307166(n).
0 <= a(n) <= A307171(n).
(End)

			For example, the square
  2 4 3 5 0 1
  1 0 4 3 2 5
  0 2 5 4 1 3
  5 3 0 1 4 2
  4 5 1 2 3 0
  3 1 2 0 5 4
has a loop
  2 4 . . . .
  . . . . . .
  . 2 . 4 . .
  . . . . . .
  4 . . 2 . .
  . . . . . .
consisting of the sequence of cells L[1,1]=2 -> L[1,2]=4 -> L[3,2]=2 -> L[3,4]=4 -> L[5,4]=2 -> L[5,1]=4 -> L[1,1]=2 with length 6 < 12.
The total number of loops for this square is 21, all of which are partial.

Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
Eduard I. Vatutin, About the minimum and maximum number of partial loops in a diagonal Latin squares of order 8 (in Russian).
Eduard I. Vatutin, On the inequalities of the minimum and maximum numerical characteristics of diagonal Latin squares for intercalates, loops and partial loops (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer (2020), 127-146.
Index entries for sequences related to Latin squares and rectangles.

Cf. A274806, A307166, A307167, A307163, A307171.

a(8) added by Eduard I. Vatutin, Oct 05 2020

A328873 Maximal size of a set of pairwise mutually orthogonal diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 2, 2, 1, 4, 6, 6

Offset: 1

Eduard I. Vatutin, Oct 29 2019

From Andrew Howroyd, Nov 08 2019: (Start)
A diagonal Latin square of order n is an n X n array with every integer from 0 to n-1 in every row, every column, and both main diagonals.
Of course if even one example exists, then a(n) >= 1.
A274806 gives the number of diagonal Latin squares and A274806(6) is nonzero. This suggests that although it is not possible to have a pair of orthogonal diagonal Latin squares, a(6) should be 1 here. (End)
a(1) = 1 because there is only one (trivial) diagonal Latin square of order 1. It is orthogonal to itself, so if we allow the consideration of multiple copies of the same diagonal Latin square, we get a(1) = infinity instead.
From Eduard I. Vatutin, Mar 27 2021: (Start)
a(n) <= A287695(n) + 1.
a(p) >= A123565(p) = p-3 for all odd prime p due to existence of clique from cyclic MODLS of order p with at least A123565(p) items. It seems that for some orders p clique from cyclic MODLS can be extended by adding none cyclic DLS that are orthogonal to all cyclic DLS. (End)
a(n) <= A001438(n). - Max Alekseyev, Nov 08 2019
a(10) >= 2; a(11) >= 8; a(12) >= 4; a(13) >= 10; a(14) >= 2; a(15) >= 4. - Natalia Makarova, Sep 03 2020; updated May 30 2021
a(16) >= 14, a(17) >= 14, a(18) >= 2, a(19) >= 16, a(20) >= 2. - Natalia Makarova, Jan 08 2021

			Orthogonal pair of Diagonal Latin squares of order 18:
   1  5 15 16 17 18  2 14  4 13  3  7 12 10  8  6 11  9
   8  2  6 15 16 17 18  1  5 14  4 13 11  9  7 12 10  3
  14  9  3  7 15 16 17  2  6  1  5 12 10  8 13 11  4 18
  13  1 10  4  8 15 16  3  7  2  6 11  9 14 12  5 18 17
  12 14  2 11  5  9 15  4  8  3  7 10  1 13  6 18 17 16
  11 13  1  3 12  6 10  5  9  4  8  2 14  7 18 17 16 15
   3 12 14  2  4 13  7  6 10  5  9  1  8 18 17 16 15 11
   9 10 11 12 13 14  1 15 16 17 18  8  7  6  5  4  3  2
   6  7  8  9 10 11 12 18 17 16 15  5  4  3  2  1 14 13
   5  6  7  8  9 10 11 16 15 18 17  4  3  2  1 14 13 12
   7  8  9 10 11 12 13 17 18 15 16  6  5  4  3  2  1 14
   4 15 16 17 18  1  8 13  3 12  2 14  6 11  9  7  5 10
  15 16 17 18 14  7  9 12  2 11  1  3 13  5 10  8  6  4
  16 17 18 13  6  8  3 11  1 10 14 15  2 12  4  9  7  5
  17 18 12  5  7  2  4 10 14  9 13 16 15  1 11  3  8  6
  18 11  4  6  1  3  5  9 13  8 12 17 16 15 14 10  2  7
  10  3  5 14  2  4  6  8 12  7 11 18 17 16 15 13  9  1
   2  4 13  1  3  5 14  7 11  6 10  9 18 17 16 15 12  8
and
   1  8 14 13 12 11  3  9  6  5  7  4 15 16 17 18 10  2
   5  2  9  1 14 13 12 10  7  6  8 15 16 17 18 11  3  4
  15  6  3 10  2  1 14 11  8  7  9 16 17 18 12  4  5 13
  16 15  7  4 11  3  2 12  9  8 10 17 18 13  5  6 14  1
  17 16 15  8  5 12  4 13 10  9 11 18 14  6  7  1  2  3
  18 17 16 15  9  6 13 14 11 10 12  1  7  8  2  3  4  5
   2 18 17 16 15 10  7  1 12 11 13  8  9  3  4  5  6 14
  14  1  2  3  4  5  6 15 16 17 18 13 12 11 10  9  8  7
   4  5  6  7  8  9 10 17 18 15 16  3  2  1 14 13 12 11
  13 14  1  2  3  4  5 18 17 16 15 12 11 10  9  8  7  6
   3  4  5  6  7  8  9 16 15 18 17  2  1 14 13 12 11 10
   7 13 12 11 10  2  1  8  5  4  6 14  3 15 16 17 18  9
  12 11 10  9  1 14  8  7  4  3  5  6 13  2 15 16 17 18
  10  9  8 14 13  7 18  6  3  2  4 11  5 12  1 15 16 17
   8  7 13 12  6 18 17  5  2  1  3  9 10  4 11 14 15 16
   6 12 11  5 18 17 16  4  1 14  2  7  8  9  3 10 13 15
  11 10  4 18 17 16 15  3 14 13  1  5  6  7  8  2  9 12
   9  3 18 17 16 15 11  2 13 12 14 10  4  5  6  7  1  8
so a(18) >= 2.

R. J. R. Abel, Charles J. Colbourn, and Jeffrey H. Dinitz, Mutually Orthogonal Latin Squares (MOLS) [Note the first author, Julian Abel, has the initials R. J. R. A. - _N. J. A. Sloane_, Nov 05 2020]
B. Du, New Bounds For Pairwise Orthogonal Diagonal Latin Squares, Australasian Journal of Combinatorics 7 (1993), pp.87-99.
Natalia Makarova, MODLS of order 15
Natalia Makarova, Complete MOLS systems
Natalia Makarova, Orthogonal Diagonal Latin squares
Natalia Makarova, Mutually Orthogonal Diagonal Latin squares (MODLS) for orders 9 - 20
Natalia Makarova, MOLS and MODLS of order 12
E. I. Vatutin, Discussion about properties of diagonal Latin squares (in Russian), Oct 29 2019.
Eduard I. Vatutin, On the falsity of Makarova's proof that a(9) = 6 (in Russian).
Eduard I. Vatutin, About the cliques from orthogonal diagonal Latin squares of order 9, brute force based proof that a(9) = 6 (in Russian).
E. I. Vatutin, M. O. Manzuk, V. S. Titov, S. E. Kochemazov, A. D. Belyshev, N. N. Nikitina, Orthogonality-based classification of diagonal latin squares of orders 1-8, High-performance computing systems and technologies. Vol. 3. No. 1. 2019. pp. 94-100. (in Russian).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, O. S. Zaikin, A. D. Belyshev, Cliques properties from diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2019). Tula, 2019. pp. 17-23. (in Russian).
Eduard I. Vatutin, About the A328873(N)-1 <= A287695(N) inequality between the maximum cardinality of clique and the maximum number of orthogonal normalized mates for one diagonal Latin square (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Wikipedia, Clique problem.
Index entries for sequences related to Latin squares and rectangles.

Cf. A001438, A274806, A287695.

a(6) corrected by Max Alekseyev and Andrew Howroyd, Nov 08 2019

a(9) added by Eduard I. Vatutin, Feb 02 2021

A305569 Number of bachelor diagonal Latin squares of order n.

Original entry on oeis.org

0, 0, 0, 0, 480, 92160, 861557760, 300261256888320, 1835082185382168791040

Offset: 1

Eduard I. Vatutin, Jun 05 2018

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

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (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)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, I. I. Citerra, Estimation of the probability of finding orthogonal diagonal Latin squares among general diagonal Latin squares, Recognition - 2018. Kursk: SWSU, 2018. pp. 72-74. (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).
Eduard I. Vatutin, Natalia N. Nikitina, Maxim O. Manzuk, Additional calculated 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).
Index entries for sequences related to Latin squares and rectangles.

Cf. A266177, A305568, A305571.

a(n) = A305568(n) * n!.

a(n) = A274806(n) - A305571(n).

a(9) added by Eduard I. Vatutin, Dec 22 2020

A307839 Minimum number of Latin subrectangles in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 137, 336, 884, 1968, 4545

Offset: 1

Eduard I. Vatutin, May 01 2019

An Latin subrectangle is a m X k Latin rectangle of a Latin square of order n, 1 <= m <= n, 1 <= k <= n.

			For example, the square
  0 1 2 3 4 5 6
  4 2 6 5 0 1 3
  3 6 1 0 5 2 4
  6 3 5 4 1 0 2
  1 5 3 2 6 4 0
  5 0 4 6 2 3 1
  2 4 0 1 3 6 5
has a Latin subrectangle
  . . . . . . .
  . . 6 5 0 1 3
  . . . . . . .
  . . . . . . .
  . . . . . . .
  . . . . . . .
  . . 0 1 3 6 5
The total number of Latin subrectangles for this square is 2119.

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
E. I. Vatutin, About the minimum and maximum number of Latin subrectangles in a diagonal Latin squares of order 8 (in Russian).
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer (2020), 127-146.
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A274806, A307163, A307164, A307840, A307841, A307842.

a(8) added by Eduard I. Vatutin, Oct 06 2020

cp's OEIS Frontend

A274171 Number of diagonal Latin squares of order n with the first row in order.

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A287648 Maximum number of diagonal transversals in a diagonal Latin square of order n.

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A305571 Number of diagonal Latin squares of order n with at least one orthogonal diagonal mate.

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A309283 Number of equivalence classes of X-based filling of diagonals in a diagonal Latin square of order n.

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A307166 Minimum number of loops in a diagonal Latin square of order n.

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A307167 Maximum number of loops in a diagonal Latin square of order n.

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A307170 Minimum number of partial loops in a diagonal Latin square of order n.

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A328873 Maximal size of a set of pairwise mutually orthogonal diagonal Latin squares of order n.

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A305569 Number of bachelor diagonal Latin squares of order n.

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