A287647 -id:A287647 - cp's OEIS frontend

A007016 Number of permutations of length n with 1 fixed and 1 reflected point.

0, 1, 0, 0, 8, 20, 96, 656, 5568, 48912, 494080, 5383552, 65097600, 840566080, 11833898496, 176621049600, 2838024476672, 48060623405312, 868000333234176, 16441638519762944, 329723762151352320, 6907027877807330304

Offset: 0

N. J. A. Sloane

Number of distinct solutions to the order n checkerboard problem, including symmetrical solutions: place n pieces on an n X n board so there is exactly one piece in each row, column and main diagonal. Compare A064280.
Number of magic permutation matrices of order n. - Chai Wah Wu, Jan 15 2019
Upper bound for the number of diagonal transversals in a Latin square: A287647(n) <= A287648(n) <= a(n). - Eduard I. Vatutin, Jan 02 2020

Simpson, Todd; Permutations with unique fixed and reflected points. Ars Combin. 39 (1995), 97-108.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jean-François Alcover, Table of n, a(n) for n = 0..100
F. Rakotondrajao, Magic squares, rook polynomials and permutations, Séminaire Lotharingien de Combinatoire, vol. 54A, article B54Ac, 2006.
T. Simpson, Letter to N. J. A. Sloane, Mar. 1992
T. Simpson, Permutations with unique fixed and reflected points, Preprint. (Annotated scanned copy)
E. Vatutin, Upper bound for the number of diagonal transversals in a Latin square (in Russian).

Cf. A003471, A064280.

x[n_] := x[n] = Integrate[If[EvenQ[n], (x^2 - 4*x + 2)^(n/2), (x - 1)*(x^2 - 4*x + 2)^((n - 1)/2)]/E^x, {x, 0, Infinity}];
a[n_ /; EvenQ[n]] := With[{m = n/2}, m*(x[2*m] - (2*m - 3)*x[2*m - 1])];
a[n_ /; OddQ[n]] := With[{m = (n - 1)/2}, (2*m + 1)*x[2*m] + 3*m*x[2*m - 1] - 2*m*(m - 1)*x[2*m - 2]];
Table[a[n], {n, 0, 21}] // Quiet (* Jean-François Alcover, Jun 29 2018 *)

a(n) = {my(v = vector(n)); \\ v is A003471
for(n=4, length(v), v[n] = (n-1)*v[n-1] + 2*if(n%2==1, (n-1)*v[n-2], (n-2) * if(n==4,1,v[n-4])));
if(n<4, [1,0,0][n], if(n%2==0, n*(v[n] - (n-3)*v[n-1]), 2*n*v[n-1] + 3*(n-1)*v[n-2] - (n-1)*(n-3)*v[n-3])/2)} \\ Andrew Howroyd, Sep 12 2017

a(2*m) = m*(x(2*m) - (2*m-3)*x(2*m-1)), a(2*m+1) = (2*m+1)*x(2*m) + 3*m*x(2*m-1) - 2*m*(m-1)*x(2*m-2), where x(n) = A003471(n).

Conjecture D-finite with recurrence (365968635435167109808*n^2 -5566069866485493251505*n +20525522573033552369132)*a(n) +(-1215369044326430542311*n^2 +19103429957352794982854*n -73690801030090785944295)*a(n-1) +(-365968635435167109808*n^4 +6663975772790994580929*n^3 -35836353442786038818589*n^2 +34878550744402035813586*n +124043542472821007763204)*a(n-2) +(483431773456096322695*n^4 -10754417727097457203127*n^3 +85154149458907095778621*n^2 -277683967994722584206067*n +286254870342835757751852)*a(n-3) +2*(-393241909113483884738*n^4 +9142334951839265043383*n^3 -78427160779754271402777*n^2 +309283968160862567580813*n -465057422344277141977923)*a(n-4) +2*(-745044547502580209919*n^4 +21471238686323774026196*n^3 -222067832543690193789255*n^2 +944698954932049830084232*n -1372732531859619119793978)*a(n-5) +4*(365968635435167109808*n^4 -5227374504728642916627*n^3 +19793104565012302929789*n^2 +391834816007939927082*n -57365695502678698166146)*a(n-6) +4*(-483431773456096322695*n^4 +7592214312314395379733*n^3 -45284933032689911393913*n^2 +117535885088909103449165*n -84799883220517633629252)*a(n-7) +8*(n-7)*(393241909113483884738*n^3 -4789400677912625536335*n^2 +17834478528905815208536*n -23668675533486426523455)*a(n-8) +8*(n-7)*(n-8)*(745044547502580209919*n^2 -6086915962816073505121*n +12854159797389104313178)*a(n-9)=0. - R. J. Mathar, Feb 27 2025

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

A287644 Maximum number of transversals in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 8, 15, 32, 133, 384, 2241

Offset: 1

Eduard I. Vatutin, May 29 2017

Same as the maximum number of transversals in a Latin square of order n except n = 3.
a(10) >= 5504 from Parker and Brown.
Every diagonal Latin square is a Latin square and every orthogonal diagonal Latin square is a diagonal Latin square, so 0 <= A287645(n) <= A357514(n) <= a(n) <= A090741(n). - Eduard I. Vatutin, added Sep 20 2020, updated Mar 03 2023
a(11) >= 37851, a(12) >= 198144, a(13) >= 1030367, a(14) >= 3477504, a(15) >= 36362925, a(16) >= 244744192, a(17) >= 1606008513, a(19) >= 87656896891, a(23) >= 452794797220965, a(25) >= 41609568918940625. - Eduard I. Vatutin, Mar 08 2020, updated Mar 10 2022
Also a(n) is the maximum number of transversals in an orthogonal diagonal Latin square of order n for all orders except n=6 where orthogonal diagonal Latin squares don't exist. - Eduard I. Vatutin, Jan 23 2022
All cyclic diagonal Latin squares are diagonal Latin squares, so A348212((n-1)/2) <= a(n) for all orders n of which cyclic diagonal Latin squares exist. - Eduard I. Vatutin, Mar 25 2021

J. W. Brown et al., Completion of the spectrum of orthogonal diagonal Latin squares, Lecture notes in pure and applied mathematics, volume 139 (1992), pp. 43-49.
E. T. Parker, Computer investigations of orthogonal Latin squares of order 10, Proc. Sympos. Appl. Math., volume 15 (1963), pp. 73-81.

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru.
E. I. Vatutin, About the minimal and maximal number of transversals in a diagonal Latin squares of order 9 (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, 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, Cham (2020), 127-146.
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).
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 (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).
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, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, On the number of transversals in diagonal Latin squares of even orders (in Russian), Cloud and distributed computing systems, within the National supercomputing forum (NSCF - 2023). Pereslavl-Zalessky, 2023. pp. 101-105.
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, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A090741, A287645, A287647, A287648, A344105, A350585, A357514.

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

a(9) added by Eduard I. Vatutin, Sep 20 2020

A287645 Minimum number of transversals in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 8, 3, 32, 7, 8, 68

Offset: 1

Eduard I. Vatutin, May 29 2017

From Eduard I. Vatutin, Sep 20 2020: (Start)
Every diagonal Latin square is a Latin square, so 0 <= a(n) <= A287644(n) <= A090741(n).
A lower bound for odd n is A091323((n-1)/2) <= a(n). (End)
By definition, the main diagonal and antidiagonal of a diagonal Latin square are transversals, so a(n)>=2 for all n>=4 (the two diagonals are the same in the order 1 square and there are no diagonal Latin squares of orders 2 or 3). - Eduard I. Vatutin, Jun 13 2021
All cyclic diagonal Latin squares are diagonal Latin squares, so a(n) <= A348212((n-1)/2) for all orders n of which cyclic diagonal Latin squares exist. - Eduard I. Vatutin, Mar 25 2021
a(10) <= 128, a(11) <= 814, a(12) <= 448, a(13) <= 43093, a(14) <= 25720, a(15) <= 215721, a(16) <= 7465984. - Eduard I. Vatutin, Mar 11 2021, updated Feb 12 2025

			From _Eduard I. Vatutin_, Apr 24 2021: (Start)
For example, diagonal Latin square
  0 1 2 3
  3 2 1 0
  1 0 3 2
  2 3 0 1
has 4 diagonal transversals (see A287648)
  0 . . .   . 1 . .   . . 2 .   . . . 3
  . . 1 .   . . . 0   3 . . .   . 2 . .
  . . . 2   . . 3 .   . 0 . .   1 . . .
  . 3 . .   2 . . .   . . . 1   . . 0 .
and 4 not diagonal transversals
  0 . . .   . 1 . .   . . 2 .   . . . 3
  . 2 . .   3 . . .   . . . 0   . . 1 .
  . . 3 .   . . . 2   1 . . .   . 0 . .
  . . . 1   . . 0 .   . 3 . .   2 . . .
total 8 transversals. (End)

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
E. I. Vatutin, About the minimal and maximal number of transversals in diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, Best examples presently known.
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)
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).
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, I. I. Kurochkin, A. M. Albertyan, On the number of transversals in diagonal Latin squares of even orders (in Russian), Cloud and distributed computing systems, within the National supercomputing forum (NSCF - 2023). Pereslavl-Zalessky, 2023. pp. 101-105.
E. Vatutin, A. Belyshev, N. Nikitina, and M. Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, Communications in Computer and Information Science, Vol. 1304, Springer, 2020, pp. 127-146, DOI: 10.1007/978-3-030-66895-2_9.
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, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, A. V. Kripachev, and A. I. Pykhtin, Methods for getting spectra of fast computable numerical characteristics of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 19-23. (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.
Index entries for sequences related to Latin squares and rectangles.

Cf. A091323, A287644, A287647, A287648, A344105.

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

a(9) added by Eduard I. Vatutin, Sep 20 2020

A345370 a(n) is the number of distinct numbers of diagonal transversals that a diagonal Latin square of order n can have.

Original entry on oeis.org

1, 0, 0, 1, 2, 2, 14, 47, 182

Offset: 1

Eduard I. Vatutin, Jun 16 2021

a(n) <= A287648(n) - A287647(n) + 1.
a(n) <= A287764(n).
Conjecture: a(12) = A287648(12) - A287647(12) + 1. - Natalia Makarova, Oct 26 2021
a(10) >= 736, a(11) >= 1344, a(12) >= 17693, a(13) >= 18241, a(14) >= 294053, a(15) >= 1958394, a(16) >= 13715. - Eduard I. Vatutin, Oct 29 2021, updated Mar 02 2025

			For n=7 the number of diagonal transversals that a diagonal Latin square of order 7 may have is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, or 27. Since there are 14 distinct values, a(7)=14.

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, On the falsity of conjecture that spectra of diagonal transversals for diagonal Latin squares of order 12 is solid (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, About the results of experiment with spectra of diagonal Latin squares using Brute Force and distributed computing projects Gerasim@Home and RakeSearch (in Russian).
Eduard I. Vatutin, Proving lists (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13).
E. I. Vatutin, Distributed diagonalization strategy for Latin squares, Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2023. pp. 309-311. (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, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, A. V. Kripachev, A. I. Pykhtin, Methods for getting spectra of fast computable numerical characteristics of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 19-23. (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)
Index entries for sequences related to Latin squares and rectangles.

Cf. A287647, A287648, A309344, A344105, A345760, A345761, A349199.

a(8) added by Eduard I. Vatutin, Jul 15 2021

a(9) added by Eduard I. Vatutin, Oct 20 2022

A342998 Minimum number of diagonal transversals in a cyclic diagonal Latin square of order 2n+1.

Original entry on oeis.org

1, 0, 5, 27, 0, 4523, 128818, 0, 204330233, 11232045257

Offset: 0

Eduard I. Vatutin, Apr 02 2021

A cyclic Latin square is a Latin square in which row i is obtained by cyclically shifting row i-1 by d places (see A338562, A123565 and A341585).
Cyclic diagonal Latin squares do not exist for even orders.
a(n) <= A342997(n).
All cyclic diagonal Latin squares are diagonal Latin squares, so A287647(n) <= a((n-1)/2).

			For n=2 one of best cyclic diagonal Latin squares 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
has a(2)=5 diagonal transversals:
  0 . . . .   . 1 . . .   . . 2 . .   . . . 3 .   . . . . 4
  . . 4 . .   . . . 0 .   . . . . 1   2 . . . .   . 3 . . .
  . . . . 3   4 . . . .   . 0 . . .   . . 1 . .   . . . 2 .
  . 2 . . .   . . 3 . .   . . . 4 .   . . . . 0   1 . . . .
  . . . 1 .   . . . . 2   3 . . . .   . 4 . . .   . . 0 . .

Eduard I. Vatutin, Enumerating the diagonal transversals for cyclic diagonal Latin squares of orders 1-19 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A006717, A123565, A287647, A287648, A338562, A341585, A342997.

A360220 Maximum number of diagonal transversals in an orthogonal diagonal Latin square of order n.

Original entry on oeis.org

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

Offset: 1

Eduard I. Vatutin, Jan 30 2023

An orthogonal diagonal Latin square is a diagonal Latin square that has at least one orthogonal diagonal mate.
a(10) >= 866, a(11) >= 4828, a(12) >= 30192, a(13) >= 131106, a(17) >= 204995269, a(19) >= 11254190082.
For most orders n, at least one diagonal Latin square with the maximal number of diagonal transversals has an orthogonal mate and A287648(n) = a(n). Known exceptions: n=6 and n=10. - Eduard I. Vatutin, Feb 17 2023
Every orthogonal diagonal Latin square is a diagonal Latin square, so A287647(n) <= A354068(n) <= a(n) <= A287648(n). - Eduard I. Vatutin, Mar 04 2023

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
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)
Index entries for sequences related to Latin squares and rectangles.

Cf. A287647, A287648, A305570, A305571, A349199, A354068.

A354068 Minimum number of diagonal transversals in an orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 4, 5, 0, 8, 8, 14

Offset: 1

Eduard I. Vatutin, May 16 2022

An orthogonal diagonal Latin square is a diagonal Latin square with at least one orthogonal diagonal mate.
a(10) <= 60, a(11) <= 279, a(12) <= 588, a(13) <= 9610.
Every orthogonal diagonal Latin square is a diagonal Latin square, so A287647(n) <= a(n) <= A360220(n) <= A287648(n). - Eduard I. Vatutin, Mar 03 2023

			One of the best orthogonal diagonal Latin squares of order n=9
  0 1 2 3 4 5 6 7 8
  1 2 3 8 6 4 7 0 5
  5 4 6 0 7 8 3 1 2
  7 3 1 5 2 6 0 8 4
  8 7 4 6 1 2 5 3 0
  3 0 5 4 8 7 1 2 6
  4 6 7 2 3 0 8 5 1
  6 5 8 1 0 3 2 4 7
  2 8 0 7 5 1 4 6 3
has orthogonal diagonal mate
  0 1 2 3 4 5 6 7 8
  2 3 8 7 5 6 4 1 0
  1 5 4 8 6 0 2 3 7
  8 7 0 6 1 3 5 4 2
  5 0 1 2 7 8 3 6 4
  4 6 7 0 3 2 8 5 1
  3 8 5 4 0 7 1 2 6
  7 4 6 5 2 1 0 8 3
  6 2 3 1 8 4 7 0 5
and 14 diagonal transversals, which is the minimal number, so a(9)=14.

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
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)
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A287647, A287648, A345370, A349199, A360220.

A387236 Minimum number of diagonal transversals in an extended self-orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 4, 5, 0, 8, 16, 15, 75

Offset: 1

Eduard I. Vatutin, Aug 23 2025

A self-orthogonal diagonal Latin square (SODLS) is a diagonal Latin square orthogonal to its transpose. An extended self-orthogonal diagonal Latin square (ESODLS) is a diagonal Latin square that has an orthogonal diagonal Latin square from the same main class. SODLS is a special case of ESODLS.

Eduard I. Vatutin, About the properties of ESODLS of orders 1-10 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A287647, A287648, A309210, A309598, A309599, A354068, A360220.

A376587 Minimum number of diagonal transversals in diagonalized cyclic diagonal Latin squares of order 2n+1.

Original entry on oeis.org

1, 0, 5, 27, 241, 4523, 127339

Offset: 0

Eduard I. Vatutin, Sep 29 2024

Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A287647, A342998, A366332, A372922, A372923, A375475.

cp's OEIS Frontend

A007016 Number of permutations of length n with 1 fixed and 1 reflected point.

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

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A287644 Maximum number of transversals in a diagonal Latin square of order n.

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A287645 Minimum number of transversals in a diagonal Latin square of order n.

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A345370 a(n) is the number of distinct numbers of diagonal transversals that a diagonal Latin square of order n can have.

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A342998 Minimum number of diagonal transversals in a cyclic diagonal Latin square of order 2n+1.

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A360220 Maximum number of diagonal transversals in an orthogonal diagonal Latin square of order n.

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A354068 Minimum number of diagonal transversals in an orthogonal diagonal Latin square of order n.

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A387236 Minimum number of diagonal transversals in an extended self-orthogonal diagonal Latin square of order n.

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